diff --git "a/naturalproofs_trench.json" "b/naturalproofs_trench.json" new file mode 100644--- /dev/null +++ "b/naturalproofs_trench.json" @@ -0,0 +1,21023 @@ +{ + "dataset": { + "theorems": [ + { + "id": 0, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.1", + "categories": [], + "title": "The Triangle Inequality", + "contents": [ + "If $a$ and $b$ are any two real numbers$,$ then", + "\\begin{equation} \\label{eq:1.1.3}", + "|a+b|\\le |a|+|b|.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "There are four possibilities:", + "\\begin{alist}", + "\\item % (a)", + "If $a\\ge0$ and $b\\ge0$, then $a+b\\ge0$, so", + "$|a+b|=a+b=|a|+|b|$.", + "\\item % (b)", + "If $a\\le0$ and $b\\le0$, then $a+b\\le0$, so", + "$|a+b|=-a+(-b)=|a|+|b|$.", + "\\item % (c)", + " If $a \\ge 0$ and $b \\le 0$, then $a+b=|a|-|b|$.", + "\\item % (d)", + " If $a \\le 0$ and $b \\ge 0$, then $a+b=-|a|+|b|$.", + "\\end{alist}", + "Eq.~\\ref{eq:1.1.3}", + "holds in cases {\\bf (c)} and {\\bf (d)}, since", + "\\begin{equation}", + "|a+b|=", + "\\begin{cases}", + "|a|-|b|& \\text{ if } |a| \\ge |b|,\\\\", + "|b|-|a|& \\text{ if } |b| \\ge |a|.", + "\\end{cases}", + "\\tag*{" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 1, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.3", + "categories": [], + "title": "", + "contents": [ + "If a nonempty set $S$ of real numbers is bounded above$,$ then", + "$\\sup S$ is the unique real number $\\beta$ such that", + "\\begin{alist}", + "\\item % (a)", + " $x\\le\\beta$ for all $x$ in $S;$", + "\\item % (b)", + " if $\\epsilon>0$ $($no matter how small$)$$,$ there is an $x_0$ in", + "$S$ such that", + "$x_0>", + "\\beta-\\epsilon.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We first show that $\\beta=\\sup S$ has properties \\part{a} and", + "\\part{b}. Since $\\beta$ is an upper bound of $S$, it must satisfy", + "\\part{a}. Since any real number $a$ less than $\\beta$ can be written", + "as $\\beta-\\epsilon$ with $\\epsilon=\\beta-a>0$, \\part{b} is just", + "another way of saying that no number less than $\\beta$ is an upper", + "bound of $S$. Hence, $\\beta=\\sup S$ satisfies \\part{a} and \\part{b}.", + "Now we show that there cannot be more than one real number with", + "properties \\part{a} and \\part{b}. Suppose that $\\beta_1<\\beta_2$ and", + "$\\beta_2$ has property \\part{b}; thus, if $\\epsilon>0$, there is an", + "$x_0$ in $S$ such that $x_0>\\beta_2-\\epsilon$. Then, by taking", + "$\\epsilon=\\beta_2-\\beta_1$, we see that there is an $x_0$ in $S$ such", + "that", + "$$", + "x_0>\\beta_2-(\\beta_2-\\beta_1)=\\beta_1,", + "$$", + "so $\\beta_1$ cannot have property \\part{a}. Therefore, there cannot", + "be more than one real number that satisfies both \\part{a} and", + "\\part{b}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 2, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.4", + "categories": [], + "title": "", + "contents": [ + "If $\\rho$ and $\\epsilon$ are positive$,$ then $n\\epsilon>\\rho$ for", + "some integer $n.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The proof is by contradiction.", + "If the statement is false, $\\rho$ is an upper bound of", + "the set", + "$$", + "S=\\set{x}{x=n\\epsilon,\\mbox{$n$ is an integer}}.", + "$$", + "Therefore, $S$ has a supremum $\\beta$, by property \\part{I}.", + "Therefore,", + "\\begin{equation}\\label{eq:1.1.9}", + "n\\epsilon\\le\\beta \\mbox{\\quad for all integers $n$}.", + "\\end{equation}}", + "\\newpage\\noindent", + "Since $n+1$ is an integer whenever $n$ is, \\eqref{eq:1.1.9} implies that", + "$$", + "(n+1)\\epsilon\\le\\beta", + "$$", + " and therefore", + "$$", + "n\\epsilon\\le\\beta-\\epsilon", + "$$", + " for all integers $n$. Hence,", + " $\\beta-\\epsilon$ is an upper bound of $S$. Since $\\beta-\\epsilon", + "<\\beta$, this contradicts the definition of~$\\beta$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 3, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.6", + "categories": [], + "title": "", + "contents": [ + "The rational numbers are dense in the reals$\\,;$ that is, if $a$", + "and", + "$b$ are real numbers with $a1$. There is also an integer", + "$j$ such that $j>qa$. This is obvious if $a\\le0$, and it follows from", + "Theorem~\\ref{thmtype:1.1.4} with $\\epsilon=1$ and $\\rho=qa$ if $a>0$. Let", + "$p$ be the smallest integer such that $p>qa$. Then $p-1\\le qa$, so", + "$$", + "qa0$ $($no matter how small$\\,)$, there is an $x_0$ in $S$", + "such that", + "$x_0<", + "\\alpha+\\epsilon.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "(Exercise~\\ref{exer:1.1.6})", + "A set $S$ is {\\it bounded\\/} if", + "there are numbers", + "$a$ and", + "$b$ such", + "that $a\\le x\\le b$ for all $x$ in $S$. A bounded nonempty set has a", + "unique supremum and a unique infimum, and", + "\\begin{equation}\\label{eq:1.1.11}", + "\\inf S\\le\\sup S", + "\\end{equation}", + "(Exercise~\\ref{exer:1.1.7})." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 6, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.1", + "categories": [], + "title": "Principle of Mathematical Induction", + "contents": [ + " Let $P_1,$ $P_2, $\\dots$,$ $P_n,$ \\dots\\ be", + "propositions$,$ one", + "for each positive integer$,$ such that", + "\\begin{alist}", + "\\item % (a)", + " $P_1$ is true$;$", + "\\item % (b)", + " for each positive integer $n,$ $P_n$ implies $P_{n+1}.$", + "\\end{alist}", + "Then $P_n$ is true for each positive integer $n.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$$", + "\\mathbb M=\\set{n}{n\\in \\mathbb N\\mbox{ and } P_n\\mbox{ is", + "true}}.", + "$$", + "From \\part{a}, $1\\in \\mathbb M$, and from \\part{b}, $n+1\\in \\mathbb M$ whenever", + "$n\\in \\mathbb M$. Therefore, $\\mathbb M=\\mathbb N$, by postulate", + "\\part{E}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 7, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.2", + "categories": [], + "title": "", + "contents": [ + " Let $n_0$ be any integer $($positive$,$", + " negative$,$ or zero$)$$.$ Let", + "$P_{n_0},$ $P_{n_0+1},$ \\dots$,$ $P_n,$ \\dots\\ be propositions$,$", + " one for each integer $n\\ge n_0,$ such that", + "\\begin{alist}", + "\\item % (a)", + " $P_{n_0}$ is true$\\,;$", + "\\item % (b)", + " for each integer $n\\ge n_0,$ $P_n$ implies $P_{n+1}.$", + "\\end{alist}", + "Then $P_n$ is true for every integer $n\\ge n_0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For $m\\ge1$, let $Q_m$ be the proposition defined by", + "$Q_m=P_{m+n_0-1}$. Then $Q_1=P_{n_0}$ is true by \\part{a}.", + "If $m\\ge1$ and $Q_m=P_{m+n_0-1}$ is true, then $Q_{m+1}=P_{m+n_0}$", + "is true by \\part{b} with $n$ replaced by $m+n_0-1$. Therefore,", + "$Q_m$ is true for all $m\\ge1$ by Theorem~\\ref{thmtype:1.2.1} with $P$", + "replaced by $Q$ and $n$ replaced by $m$. This is equivalent", + "to the statement that $P_n$ is true for all $n\\ge n_0$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.2.1" + ], + "ref_ids": [ + 6 + ] + } + ], + "ref_ids": [] + }, + { + "id": 8, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.2.3", + "categories": [], + "title": "", + "contents": [ + " Let $n_0$ be any integer $($positive$,$", + " negative$,$ or zero$)$$.$ Let", + "$P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n,$ \\dots\\ be propositions$,$", + " one for each integer $n\\ge n_0,$ such that", + "\\begin{alist}", + "\\item % (a)", + " $P_{n_0}$ is true$\\,;$", + "\\item % (b)", + "for $n\\ge n_0,$ $P_{n+1}$ is true if $P_{n_0},$ $P_{n_0+1}, $\\dots$,$", + "$P_n$ are all true.", + "\\end{alist}", + "Then $P_n$ is true for $n\\ge n_0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For $n\\ge n_0$, let $Q_n$ be the proposition that", + " $P_{n_0}$, $P_{n_0+1}$, \\dots, $P_n$ are all true.", + "Then $Q_{n_0}$ is true by \\part{a}. Since $Q_n$ implies $P_{n+1}$", + "by \\part{b}, and $Q_{n+1}$ is true if $Q_n$ and $P_{n+1}$ are both true,", + "Theorem~\\ref{thmtype:1.2.2} implies that $Q_n$ is true for all $n\\ge", + "n_0$. Therefore, $P_n$ is true for all $n\\ge n_0$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.2.2" + ], + "ref_ids": [ + 7 + ] + } + ], + "ref_ids": [] + }, + { + "id": 9, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.3", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + " The union of open sets is open$.$", + "\\item % (b)", + " The intersection of closed sets is closed$.$", + "\\end{alist}", + "These statements apply to", + "arbitrary collections, finite or infinite, of open and closed", + "sets$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} Let ${\\mathcal G}$ be a collection of open sets and", + "$$", + "S=\\cup\\set{G}{G\\in {\\mathcal G}}.", + "$$", + "If $x_0\\in S$, then $x_0\\in G_0$ for some $G_0$ in ${\\mathcal G}$, and", + "since $G_0$ is open, it contains some $\\epsilon$-neighborhood of", + "$x_0$. Since $G_0\\subset S$, this $\\epsilon$-neighborhood is in $S$,", + "which is consequently a neighborhood of $x_0$. Thus, $S$ is a", + "neighborhood of each of its points, and therefore open, by definition.", + "\\part{b} Let ${\\mathcal F}$ be a collection of closed sets and $T", + "=\\cap\\set{F}{F\\in {\\mathcal F}}$. Then $T^c=\\cup\\set{F^c}{F\\in {\\mathcal", + "F}}$", + "(Exercise~\\ref{exer:1.3.7}) and, since each $F^c$ is open,", + "$T^c$ is open, from \\part{a}. Therefore, $T$ is closed, by", + "definition." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 10, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.5", + "categories": [], + "title": "", + "contents": [ + "no point of $S^c$ is a limit point of~$S.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $S$ is closed and $x_0\\in S^c$. Since $S^c$ is open,", + "there is a neighborhood of $x_0$ that is contained in $S^c$ and", + "therefore contains no points of $S$. Hence, $x_0$ cannot be a limit", + "point of $S$. For the converse, if no point of $S^c$ is a limit point", + "of $S$ then every point in $S^c$ must have a neighborhood contained", + "in $S^c$. Therefore, $S^c$ is open and $S$ is closed." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 11, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.7", + "categories": [], + "title": "", + "contents": [ + "If ${\\mathcal H}$ is an open covering of a closed and bounded subset $S$", + "of the real line$,$ then $S$ has an open covering $\\widetilde{\\mathcal", + "H}$ consisting of finitely many open sets belonging to ${\\mathcal H}.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $S$ is bounded, it has an infimum $\\alpha$", + "and a supremum $\\beta$, and, since $S$ is closed, $\\alpha$", + "and $\\beta$ belong to $S$ (Exercise~\\ref{exer:1.3.17}). Define", + "$$", + "S_t=S\\cap [\\alpha,t] \\mbox{\\quad for \\ } t\\ge\\alpha,", + "$$", + "and let", + "$$", + "F=\\set{t}{\\alpha\\le t\\le\\beta \\mbox{\\ and finitely many sets from", + "${\\mathcal H}$ cover $S_t$}}.", + "$$", + "Since $S_\\beta=S$, the theorem will be proved if we can show that", + "$\\beta", + "\\in F$. To do this, we use the completeness of the reals.", + "Since $\\alpha\\in S$, $S_\\alpha$ is the singleton set $\\{\\alpha\\}$,", + "which is contained in some open set $H_\\alpha$ from ${\\mathcal H}$", + "because ${\\mathcal H}$ covers $S$; therefore, $\\alpha\\in F$. Since $F$ is", + "nonempty and bounded above by $\\beta$, it has a supremum $\\gamma$.", + "First, we wish to show that $\\gamma=\\beta$. Since $\\gamma\\le\\beta$ by", + "definition of $F$, it suffices to rule out the possibility that", + "$\\gamma<\\beta$. We consider two cases.", + "{\\sc Case 1}. Suppose that $\\gamma<\\beta$ and $\\gamma\\not\\in S$. Then,", + "since $S$ is closed, $\\gamma$ is not a limit point of $S$", + "(Theorem~\\ref{thmtype:1.3.5}). Consequently, there is an $\\epsilon>0$", + "such that", + "$$", + "[\\gamma-\\epsilon,\\gamma+\\epsilon]\\cap S=\\emptyset,", + "$$", + "so $S_{\\gamma-\\epsilon}=S_{\\gamma+\\epsilon}$. However, the", + "definition of $\\gamma$ implies that $S_{\\gamma-\\epsilon}$ has a finite", + "subcovering from ${\\mathcal H}$, while $S_{\\gamma+\\epsilon}$ does not.", + "This is a contradiction.", + "{\\sc Case 2}. Suppose that $\\gamma<\\beta$ and $\\gamma\\in S$. Then", + "there is an open", + "set $H_\\gamma$ in ${\\mathcal H}$ that contains $\\gamma$ and, along with $\\gamma$, an", + "interval $[\\gamma-\\epsilon,\\gamma+\\epsilon]$ for some positive", + "$\\epsilon$.", + "Since $S_{\\gamma-\\epsilon}$ has a finite covering $\\{H_1, \\dots,H_n\\}$ of", + "sets from ${\\mathcal H}$, it follows that $S_{\\gamma+\\epsilon}$ has the finite", + "covering $\\{H_1, \\dots,H_n,H_\\gamma\\}$. This contradicts the", + "definition of $\\gamma$.", + "Now we know that $\\gamma=\\beta$, which is in $S$. Therefore, there is", + "an open set $H_\\beta$ in ${\\mathcal H}$ that contains $\\beta$ and along", + "with $\\beta$, an interval of the form", + "$[\\beta-\\epsilon,\\beta+\\epsilon]$, for some positive $\\epsilon$. Since", + "$S_{\\beta-\\epsilon}$ is covered by a finite collection of sets", + "$\\{H_1, \\dots,H_k\\}$, $S_\\beta$ is covered by the finite collection", + "$\\{H_1, \\dots, H_k, H_\\beta\\}$. Since $S_\\beta=S$, we are", + "finished." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.3.5" + ], + "ref_ids": [ + 10 + ] + } + ], + "ref_ids": [] + }, + { + "id": 12, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.8", + "categories": [], + "title": "", + "contents": [ + " Every bounded infinite set of real numbers has at least one", + "limit point$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will show that a bounded nonempty set without a limit point", + "can contain only a finite number of points. If $S$ has no limit", + "points, then $S$ is closed (Theorem~\\ref{thmtype:1.3.5}) and every point", + "$x$ of $S$ has an open neighborhood $N_x$ that contains no point of", + "$S$ other than $x$. The collection", + "$$", + "{\\mathcal H}=\\set{N_x}{x\\in S}", + "$$", + "is an open covering for $S$. Since $S$ is also bounded,", + "Theorem~\\ref{thmtype:1.3.7} implies that $S$ can be covered by a finite", + "collection of sets from ${\\mathcal H}$, say $N_{x_1}$, \\dots, $N_{x_n}$.", + "Since", + "these sets contain only $x_1$, \\dots, $x_n$ from $S$, it follows that", + "$S=\\{x_1, \\dots,x_n\\}$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:1.3.7" + ], + "ref_ids": [ + 10, + 11 + ] + } + ], + "ref_ids": [] + }, + { + "id": 13, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.3", + "categories": [], + "title": "", + "contents": [ + "then it is unique$\\,;$ that is$,$ if", + "\\begin{equation} \\label{eq:2.1.7}", + "\\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} f(x)=", + "L_2,", + "\\end{equation}", + "then $L_1=L_2.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that \\eqref{eq:2.1.7} holds and let $\\epsilon>0$.", + "From Definition~\\ref{thmtype:2.1.2}, there are", + "positive numbers $\\delta_1$ and $\\delta_2$ such that", + "$$", + "|f(x)-L_i|<\\epsilon\\mbox{\\quad if \\quad} 0<|x-x_0|<\\delta_i,", + "\\quad i=1,2.", + "$$", + "If $\\delta=\\min(\\delta_1,\\delta_2)$, then", + "\\begin{eqnarray*}", + "|L_1-L_2|\\ar= |L_1-f(x)+f(x)-L_2|\\\\", + "\\ar \\le|L_1-f(x)|+|f(x)-L_2|<2\\epsilon", + "\\mbox{\\quad if \\quad} 0<|x-x_0|<\\delta.", + "\\end{eqnarray*}", + "We have now established an inequality that does not depend on $x$;", + "that is,", + "$$", + "|L_1-L_2|<2\\epsilon.", + "$$", + "Since this holds for any positive $\\epsilon$,", + " $L_1=L_2$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.1.2" + ], + "ref_ids": [ + 303 + ] + } + ], + "ref_ids": [] + }, + { + "id": 14, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.4", + "categories": [], + "title": "", + "contents": [ + "\\begin{equation}\\label{eq:2.1.9}", + "\\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} g(x)=", + "L_2,", + "\\end{equation}", + "then", + "\\begin{eqnarray}", + "\\lim_{x\\to x_0} (f+g)(x)\\ar= L_1+L_2,\\label{eq:2.1.10}\\\\", + "\\lim_{x\\to x_0} (f-g)(x)\\ar= L_1-L_2,\\label{eq:2.1.11}\\\\", + "\\lim_{x\\to x_0} (fg)(x)\\ar= L_1L_2,\\label{eq:2.1.12}\\\\", + "\\arraytext{and, if $L_2\\ne0$,}\\\\", + "\\lim_{x\\to x_0}\\left(\\frac{f}{g}\\right)(x)\\ar= \\frac{L_1}{", + "L_2}.\\label{eq:2.1.13}", + "\\end{eqnarray}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From \\eqref{eq:2.1.9} and Definition~\\ref{thmtype:2.1.2},", + " if $\\epsilon>0$, there is a", + "$\\delta_1>0$ such that", + "\\begin{equation}\\label{eq:2.1.14}", + "|f(x)-L_1|<\\epsilon", + "\\end{equation}", + "if $0<|x-x_0|<\\delta_1$, and a $\\delta_2>0$ such that", + "\\begin{equation}\\label{eq:2.1.15}", + "|g(x)-L_2|<\\epsilon", + "\\end{equation}", + "if $0<|x-x_0|<\\delta_2$. Suppose that", + "\\begin{equation}\\label{eq:2.1.16}", + "0<|x-x_0|<\\delta=\\min (\\delta_1,\\delta_2),", + "\\end{equation}", + "so that \\eqref{eq:2.1.14} and \\eqref{eq:2.1.15} both hold. Then", + "\\begin{eqnarray*}", + "|(f\\pm g)(x)-(L_1\\pm L_2)|\\ar= |(f(x)-L_1)\\pm", + "(g(x)-L_2)|\\\\", + "\\ar \\le|f(x)-L_1|+|g(x)-L_2|<2\\epsilon,", + "\\end{eqnarray*}", + "which proves \\eqref{eq:2.1.10} and \\eqref{eq:2.1.11}.", + "To prove \\eqref{eq:2.1.12}, we assume \\eqref{eq:2.1.16} and write", + "\\begin{eqnarray*}", + "|(fg)(x)-L_1L_2|\\ar= |f(x)g(x)-L_1L_2|\\\\[.5\\jot]", + "\\ar= |f(x)(g(x)-L_2)+L_2(f(x)-L_1)|\\\\[.5\\jot]", + "\\ar \\le|f(x)||g(x)-L_2|+|L_2||f(x)-L_1|\\\\[.5\\jot]", + "\\ar \\le(|f(x)|+|L_2|)\\epsilon\\mbox{\\quad (from \\eqref{eq:2.1.14} and", + "\\eqref{eq:2.1.15})}\\\\[.5\\jot]", + "\\ar \\le(|f(x)-L_1|+|L_1|+|L_2|)\\epsilon\\\\[.5\\jot]", + "\\ar \\le(\\epsilon+|L_1|+|L_2|)\\epsilon\\mbox{\\quad from", + "\\eqref{eq:2.1.14}}\\\\[.5\\jot]", + "\\ar \\le (1+|L_1|+|L_2|)\\epsilon", + "\\end{eqnarray*}", + "if $\\epsilon<1$", + "and $x$ satisfies \\eqref{eq:2.1.16}. This proves", + "\\eqref{eq:2.1.12}.", + "To prove \\eqref{eq:2.1.13}, we first observe that if $L_2\\ne0$, there is", + "a $\\delta_3>0$ such that", + "$$", + "|g(x)-L_2|<\\frac{|L_2|}{2},", + "$$", + "so", + "\\begin{equation} \\label{eq:2.1.17}", + "|g(x)|>\\frac{|L_2|}{2}", + "\\end{equation}", + "if", + "$$", + "0<|x-x_0|<\\delta_3.", + "$$", + "To see this, let $L=L_2$ and $\\epsilon=|L_2|/2$ in", + "\\eqref{eq:2.1.4}. Now suppose that", + "$$", + "0<|x-x_0|<\\min", + "(\\delta_1,\\delta_2,\\delta_3),", + "$$", + "\\nopagebreak", + " so that \\eqref{eq:2.1.14}, \\eqref{eq:2.1.15},", + "and \\eqref{eq:2.1.17} all hold. Then", + "\\pagebreak", + "\\begin{eqnarray*}", + "\\left|\\left(\\frac{f}{ g}\\right)(x)-\\frac{L_1}{ L_2}\\right|", + "\\ar= \\left|\\frac{f(x)}{ g(x)}-\\frac{L_1}{ L_2}\\right|\\\\", + "\\ar= \\frac{|L_2f(x)-L_1g(x)|}{|g(x)L_2|}\\\\", + "\\ar \\le\\frac{2}{ |L_2|^2}|L_2f(x)-L_1g(x)|\\\\", + "\\ar= \\frac{2}{ |L_2|^2}\\left|L_2[f(x)-L_1]+", + "L_1[L_2-g(x)]\\right|\\mbox{\\quad (from \\eqref{eq:2.1.17})}\\\\", + "\\ar \\le\\frac{2}{ |L_2|^2}\\left[|L_2||f(x)-L_1|+|L_1|", + "|L_2-g(x)|\\right]\\\\", + "\\ar \\le\\frac{2}{ |L_2|^2}(|L_2|+|L_1|)\\epsilon", + "\\mbox{\\quad (from \\eqref{eq:2.1.14} and \\eqref{eq:2.1.15})}.", + "\\end{eqnarray*}", + "This proves \\eqref{eq:2.1.13}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.1.2" + ], + "ref_ids": [ + 303 + ] + } + ], + "ref_ids": [] + }, + { + "id": 15, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.6", + "categories": [], + "title": "", + "contents": [ + "A function $f$ has a limit at $x_0$", + "if and only if it has left- and right-hand limits at $x_0,$ and they", + "are equal. More specifically$,$", + "$$", + "\\lim_{x\\to x_0} f(x)=L", + "$$", + "if and only if", + "$$", + "f(x_0+)=f(x_0-)=L.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 16, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.9", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is monotonic on $(a,b)$ and define", + "$$", + "\\alpha=\\inf_{a\\alpha$, there is an $x_0$ in $(a,b)$ such that $f(x_0)-\\infty$, let", + "$M=\\alpha+\\epsilon$, where $\\epsilon>0$. Then $\\alpha\\le", + "f(x)<\\alpha+\\epsilon$, so", + "\\begin{equation} \\label{eq:2.1.20}", + "|f(x)-\\alpha|<\\epsilon\\mbox{\\quad if \\quad} a-\\infty$, let $\\delta=x_0-a$. Then \\eqref{eq:2.1.20} is equivalent to", + "$$", + "|f(x)-\\alpha|<\\epsilon\\mbox{\\quad if \\quad} aM$. Since $f$ is nondecreasing, $f(x)>M$ if", + "$x_00$. Then", + "$\\beta-\\epsilon< f(x)\\le\\beta$, so", + "\\begin{equation} \\label{eq:2.1.21}", + "|f(x)-\\beta|<\\epsilon\\mbox{\\quad if \\quad} x_00$, there is an $a_1$ in $[a,x_0)$ such that", + "\\begin{equation} \\label{eq:2.1.22}", + "f(x)<\\beta+\\epsilon\\mbox{\\quad if \\quad} a_1\\le x0$ and $a_1$ is in $[a,x_0),$ then", + "$$", + "f(\\overline x)>\\beta-\\epsilon\\mbox{\\quad for some }\\overline", + "x\\in[a_1,x_0).", + "$$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $f$ is bounded on $[a,x_0)$, $S_f(x;x_0)$ is nonincreasing and", + "bounded on $[a,x_0)$. By applying Theorem~\\ref{thmtype:2.1.9}\\part{b} to", + "$S_f(x;x_0)$, we conclude that $\\beta$", + "exists (finite). Therefore, if $\\epsilon>0$, there is an $\\overline a$", + "in", + "$[a,x_0)$ such that}", + "\\begin{equation} \\label{eq:2.1.23}", + "\\beta-\\epsilon/2\\beta-\\epsilon/2.", + "\\end{equation}", + "Since $S_f(x_1;x_0)$ is the supremum of $\\set{f(t)}{x_1S_f(x_1;x_0)-\\epsilon/2.", + "$$", + "This and \\eqref{eq:2.1.24} imply that $f(\\overline x)>\\beta-\\epsilon$.", + "Since $\\overline x$ is in $[a_1,x_0)$, this proves \\part{b}.", + "Now we show that there cannot be more than one real number with", + "properties \\part{a} and \\part{b}. Suppose that $\\beta_1<\\beta_2$ and", + "$\\beta_2$ has property \\part{b}; thus, if $\\epsilon>0$ and $a_1$ is", + "in $[a,x_0)$, there is an", + "$\\overline x$ in $[a_1,x_0)$ such that", + "$f(\\overline x)>\\beta_2-\\epsilon$. Letting", + "$\\epsilon=\\beta_2-\\beta_1$, we see that there is an $\\overline x$ in", + " $[a_1,b)$ such that", + "$$", + "f(\\overline x)>\\beta_2-(\\beta_2-\\beta_1)=\\beta_1,", + "$$", + "so $\\beta_1$ cannot have property \\part{a}. Therefore, there cannot", + "be more than one real number that satisfies both \\part{a} and", + "\\part{b}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.1.9" + ], + "ref_ids": [ + 16 + ] + } + ], + "ref_ids": [] + }, + { + "id": 18, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.12", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on $[a,x_0),$", + "then $\\alpha=\\liminf_{x\\to x_0-}f(x)$ exists", + "and is the unique real number with the following properties:", + "\\begin{alist}", + "\\item % (a)", + "If $\\epsilon>0,$ there is an $a_1$ in $[a,x_0)$ such that", + "$$", + "f(x)>\\alpha-\\epsilon\\mbox{\\quad if \\quad} a_1\\le x0$ and $a_1$ is in $[a,x_0),$ then", + "$$", + "f(\\overline x)<\\alpha+\\epsilon\\mbox{\\quad for some }\\overline", + "x\\in[a_1,x_0).", + "$$", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 19, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.2", + "categories": [], + "title": "", + "contents": [ + "\\vspace*{6pt}", + "\\begin{alist}", + "\\item % (a)", + "A function $f$ is continuous at $x_0$ if and only if $f$ is defined on", + "an open interval $(a,b)$ containing $x_0$ and for each", + "$\\epsilon>0$ there is a $\\delta >0$ such that", + "\\begin{equation}\\label{eq:2.2.1}", + "|f(x)-f(x_0)|<\\epsilon", + "\\end{equation}", + "whenever $|x-x_0|<\\delta.$", + "\\item % (b)", + "A function $f$ is continuous from the right at $x_0$ if and only if", + "$f$ is defined on an interval $[x_0,b)$ and for each $\\epsilon>", + "0$", + "there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever $x_0\\le", + "x0$", + "there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever", + "$x_0-\\delta0$. Since $g(x_0)$ is an interior", + "point of $D_f$ and $f$ is continuous at $g(x_0)$, there is a", + "$\\delta_1>0$ such that $f(t)$ is defined and", + "\\begin{equation}\\label{eq:2.2.4}", + "|f(t)-f(g(x_0))|<\\epsilon\\mbox{\\quad if \\quad} |t-g(x_0)|<", + "\\delta_1.", + "\\end{equation}", + "Since $g$ is continuous at $x_0$, there is a $\\delta>0$ such that", + "$g(x)$ is defined and", + "\\begin{equation}\\label{eq:2.2.5}", + "|g(x)-g(x_0)|<\\delta_1\\mbox{\\quad if \\quad}|x-x_0|<\\delta.", + "\\end{equation}", + "Now \\eqref{eq:2.2.4} and \\eqref{eq:2.2.5} imply that", + "$$", + "|f(g(x))-f(g(x_0))|<\\epsilon\\mbox{\\quad if \\quad}|x-x_0|<\\delta.", + "$$", + " Therefore, $f\\circ g$ is continuous at $x_0$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 22, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a finite closed interval $[a,b],$ then $f$ is", + "bounded on $[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $t\\in [a,b]$. Since $f$ is continuous at $t$,", + "there is an open interval $I_t$ containing $t$ such", + "that", + "\\begin{equation}\\label{eq:2.2.7}", + "|f(x)-f(t)|<1 \\mbox{\\quad if \\quad}\\ x\\in I_t\\cap [a,b].", + "\\end{equation}", + "(To see this, set $\\epsilon=1$ in \\eqref{eq:2.2.1},", + "Theorem~\\ref{thmtype:2.2.2}.) The collection", + "${\\mathcal H}=\\set{I_t}{a\\le t\\le b}$", + "is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", + "Heine--Borel theorem implies that there are finitely many points", + "$t_1$,", + "$t_2$, \\dots, $t_n$ such that the intervals $I_{t_1}$,", + "$I_{t_2}$, \\dots, $I_{t_n}$", + "cover $[a,b]$. According to \\eqref{eq:2.2.7} with $t=t_i$,", + "$$", + "|f(x)-f(t_i)|<1\\mbox{\\quad if \\quad}\\ x\\in I_{t_i}\\cap [a,b].", + "$$", + "Therefore,", + "\\begin{equation}\\label{eq:2.2.8}", + "\\begin{array}{rcl}", + "|f(x)|\\ar =|(f(x)-f(t_i))+f(t_i)|\\le|f(x)-f(t_i)|+|f(t_i)|\\\\[2\\jot]", + "\\ar\\le 1+|f(t_i)|\\mbox{\\quad if \\quad}\\", + "x\\in I_{t_i}\\cap[a,b].", + "\\end{array}", + "\\end{equation}", + " Let", + "$$", + "M=1+\\max_{1\\le i\\le n}|f(t_i)|.", + "$$", + "Since $[a,b]\\subset\\bigcup^n_{i=1}\\left(I_{t_i}\\cap", + "[a,b]\\right)$, \\eqref{eq:2.2.8} implies that", + "$|f(x)|\\le M$ if $x\\in [a,b]$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.2.2" + ], + "ref_ids": [ + 19 + ] + } + ], + "ref_ids": [] + }, + { + "id": 23, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is continuous on a finite closed interval $[a,b].$ Let", + "$$", + "\\alpha=\\inf_{a\\le x\\le b}f(x)\\mbox{\\quad and", + "\\quad}\\beta=\\sup_{a\\le x\\le b}f(x).", + "$$", + "Then $\\alpha$ and $\\beta$ are respectively the minimum", + "and maximum of $f$ on $[a,b];$ that is$,$", + " there are points $x_1$ and $x_2$ in $[a,b]$ such that", + "$$", + "f(x_1)=\\alpha\\mbox{\\quad and \\quad} f(x_2)=\\beta.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We show that $x_1$ exists and leave it to you to show that $x_2$", + "exists (Exercise~\\ref{exer:2.2.24}).", + "Suppose that there is no", + "$x_1$ in $[a,b]$ such that $f(x_1)=\\alpha$. Then $f(x)>\\alpha$", + "for all $x\\in[a,b]$. We will show that this leads to a", + "contradiction.", + "Suppose that $t\\in[a,b]$.", + "Then $f(t)>\\alpha$, so", + "$$", + "f(t)>\\frac{f(t)+\\alpha}{2}>\\alpha.", + "$$", + "\\enlargethispage{1in}", + "\\newpage", + "\\noindent", + "Since $f$ is continuous at $t$, there is an open interval $I_t$ about", + "$t$ such that", + "\\begin{equation}\\label{eq:2.2.9}", + "f(x)>\\frac{f(t)+\\alpha}{2}\\mbox{\\quad if \\quad} x\\in", + "I_t\\cap [a,b]", + "\\end{equation}", + "(Exercise~\\ref{exer:2.2.15}). The collection ${\\mathcal H}=\\set{I_t}{a\\le t\\le", + "b}$ is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", + "Heine--Borel theorem implies that there are finitely many points $t_1$,", + "$t_2$, \\dots, $t_n$ such that the intervals $I_{t_1}$,", + "$I_{t_2}$, \\dots,", + "$I_{t_n}$ cover $[a,b]$. Define", + "$$", + "\\alpha_1=\\min_{1\\le i\\le n}\\frac{f(t_i)+\\alpha}{2}.", + "$$", + "Then, since $[a,b]\\subset\\bigcup^n_{i=1} (I_{t_i}\\cap [a,b])$,", + "\\eqref{eq:2.2.9} implies that", + "$$", + "f(t)>\\alpha_1,\\quad a\\le t\\le b.", + "$$", + "But $\\alpha_1>\\alpha$, so this contradicts the definition of $\\alpha$.", + "Therefore, $f(x_1)=\\alpha$ for some $x_1$ in $[a,b]$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 24, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.10", + "categories": [], + "title": "Intermediate Value Theorem", + "contents": [ + "Suppose that $f$ is continuous on $[a,b],$ $f(a)\\ne f(b),$ and $\\mu$", + "is between $f(a)$ and $f(b).$ Then $f(c)=\\mu$ for some", + "$c$ in $(a,b).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $f(a)<\\mu\\mu$, then $c>a$ and, since $f$ is", + "continuous at $c$, there is an $\\epsilon>0$ such that", + "$f(x)>\\mu$ if $c-\\epsilon0$ such that $f(x)<\\mu$ for $c\\le", + "x0$. Since $f$ is continuous on $[a,b]$,", + "for each $t$ in $[a,b]$ there is a positive number", + "$\\delta_{t}$ such that", + "\\begin{equation}\\label{eq:2.2.10}", + "|f(x)-f(t)|<\\frac{\\epsilon}{2}", + "\\mbox{\\quad if \\quad}", + "|x-t|<2\\delta_{t}", + "\\mbox{\\quad and \\quad} x\\in[a,b].", + "\\end{equation}", + "If $I_{t}=(t-\\delta_{t", + "},t+\\delta_{t})$, the collection", + "$$", + "{\\mathcal H}=\\set{I_{t}}{t\\in [a,b]}", + "$$", + "is an open covering of $[a,b]$. Since $[a,b]$ is compact, the", + "Heine--Borel theorem implies that there are finitely many points", + "$t_1$, $t_2$, \\dots, $t_n$ in", + "$[a,b]$ such that $I_{t_1}$, $I_{t_2}$, \\dots, $I_{t_n}$ cover", + "$[a,b]$. Now define", + "\\begin{equation}\\label{eq:2.2.11}", + "\\delta=\\min\\{\\delta_{t_1},\\delta_{t_2}, \\dots,\\delta_{t_n}\\}.", + "\\end{equation}", + "We will show that if", + "\\begin{equation} \\label{eq:2.2.12}", + "|x-x'|<\\delta \\mbox{\\quad and \\quad}x,x'\\in [a,b],", + "\\end{equation}", + "then", + "$|f(x)-f(x')|<\\epsilon$.", + "From the triangle inequality,", + "\\begin{equation} \\label{eq:2.2.13}", + "\\begin{array}{rcl}", + "|f(x)-f(x')|\\ar =", + "|\\left(f(x)-f(t_r)\\right)+\\left(f(t_r)-f(x')\\right)|\\\\", + "\\ar\\le |f(x)-f(t_r)|+|f(t_r)-f(x')|.", + "\\end{array}", + "\\end{equation}", + "Since $I_{t_1}$, $I_{t_2}$, \\dots, $I_{t_n}$ cover $[a,b]$, $x$ must", + "be in one of", + "these intervals. Suppose that", + "$x\\in I_{t_r}$; that is,", + "\\begin{equation} \\label{eq:2.2.14}", + "|x-t_r|<\\delta_{t_r}.", + "\\end{equation}", + "From \\eqref{eq:2.2.10} with $t=t_r$,", + "\\begin{equation} \\label{eq:2.2.15}", + "|f(x)-f(t_r)|<\\frac{\\epsilon}{2}.", + "\\end{equation}", + "From \\eqref{eq:2.2.12}, \\eqref{eq:2.2.14}, and the triangle inquality,", + "$$", + "|x'-t_r|=|(x'-x)+(x-t_r)|\\le", + " |x'-x|+|x-t_r|<\\delta+\\delta_{t_r}\\le2\\delta_{t_r}.", + "$$", + "Therefore, \\eqref{eq:2.2.10} with $t=t_r$ and $x$ replaced by", + "$x'$ implies that", + "$$", + "|f(x')-f(t_r)|<\\frac{\\epsilon}{2}.", + "$$", + "This, \\eqref{eq:2.2.13}, and \\eqref{eq:2.2.15} imply that", + "$|f(x)-f(x')|<\\epsilon$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 26, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.14", + "categories": [], + "title": "", + "contents": [ + "If $f$ is monotonic and nonconstant on $[a,b],$ then $f$ is continuous", + "on $[a,b]$ if and only if its range $R_f=\\set{f(x)}{x\\in[a,b]}$ is the", + "closed interval with endpoints $f(a)$ and $f(b).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We assume that $f$ is nondecreasing, and", + "leave the case where $f$ is nonincreasing to you", + "(Exercise~\\ref{exer:2.2.34}).", + "Theorem~\\ref{thmtype:2.1.9}\\part{a}", + "implies that the set $\\widetilde R_f=\\set{f(x)}{x\\in(a,b)}$", + "is a subset of the open interval $(f(a+),f(b-))$. Therefore,", + "\\begin{equation} \\label{eq:2.2.16}", + "R_f=\\{f(a)\\}\\cup\\widetilde", + "R_f\\cup\\{f(b)\\}\\subset\\{f(a)\\}\\cup(f(a+),f(b-))\\cup\\{f(b)\\}.", + "\\end{equation}", + "Now", + "suppose that $f$ is continuous on $[a,b]$. Then $f(a)=f(a+)$,", + "$f(b-)=f(b)$,", + "so \\eqref{eq:2.2.16} implies that", + "$R_f\\subset[f(a),f(b)]$. If $f(a)<\\mu0$ such that", + "$$", + "|E(x)|<|f'(x_0)|\\mbox{\\quad if\\quad} |x-x_0|<\\delta,", + "$$", + "and the right side of \\eqref{eq:2.3.16} must have the same sign as", + "$f'(x_0)$ for $|x-x_0|<\\delta$. Since the same is true of the left", + "side, $f(x)-f(x_0)$ must change sign in every neighborhood of $x_0$", + "(since $x-x_0$ does). Therefore, neither \\eqref{eq:2.3.14} nor", + "\\eqref{eq:2.3.15} can hold for all $x$ in any interval about $x_0$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.3.2" + ], + "ref_ids": [ + 244 + ] + } + ], + "ref_ids": [] + }, + { + "id": 32, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.8", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is continuous on the closed interval $[a,b]$ and", + "differentiable on the open interval $(a,b),$ and $f(a)=f(b).$ Then", + "$f'(c)=0$ for some $c$ in the open interval $(a,b).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $f$ is continuous on $[a,b]$, $f$ attains a maximum and a", + "minimum", + "value on $[a,b]$ (Theorem~\\ref{thmtype:2.2.9}). If these two", + "extreme values are the same, then $f$ is constant on $(a,b)$, so", + "$f'(x)=0$ for all $x$ in $(a,b)$. If the extreme values differ, then", + "at least one must be attained at some point $c$ in the open interval", + "$(a,b)$, and $f'(c)=0$, by Theorem~\\ref{thmtype:2.3.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", + "TRENCH_REAL_ANALYSIS-thmtype:2.3.7" + ], + "ref_ids": [ + 23, + 31 + ] + } + ], + "ref_ids": [] + }, + { + "id": 33, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.9", + "categories": [], + "title": "Intermediate Value Theorem for Derivatives", + "contents": [ + " Suppose that $f$ is differentiable on $[a,b],$ $f'(a)\\ne", + "f'(b),$ and $\\mu$ is between $f'(a)$ and $f'(b).$ Then $f'(c)=\\mu$", + "for some $c$ in $(a,b).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose first that", + "\\begin{equation}\\label{eq:2.3.17}", + "f'(a)<\\mu0.", + "\\end{equation}", + "Since $g$ is", + "continuous on $[a,b]$, $g$ attains a minimum at some point $c$ in", + "$[a,b]$. Lemma~\\ref{thmtype:2.3.2} and \\eqref{eq:2.3.19} imply that there is a", + "$\\delta>0$ such that", + "$$", + "g(x)0,\\quad f'(x)\\ge0,\\quad f'(x)<0,\\mbox{\\quad or\\quad} f'(x)", + "\\le0,", + "$$", + "respectively$,$ for all $x$ in $(a,b).$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 38, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.14", + "categories": [], + "title": "", + "contents": [ + "If", + "$$", + "|f'(x)|\\le M,\\quad a0$. From \\eqref{eq:2.4.3}, there is an $x_0$ in $(a,b)$ such", + "that", + "\\begin{equation}\\label{eq:2.4.5}", + "\\left|\\frac{f'(c)}{g'(c)}-L\\right|<\\epsilon\\mbox{\\quad if\\quad}", + "x_0", + "x_0$", + "so that $f(x)\\ne0$ and $f(x)\\ne f(x_0)$ if $x_10.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $f^{(r)}(x_0)=0$ for $1\\le r\\le n-1$,", + " \\eqref{eq:2.5.7} implies that", + "\\begin{equation}\\label{eq:2.5.10}", + "f(x)-f(x_0)=\\left[\\frac{f^{(n)}(x_0)}{ n!}+E_n(x)\\right] (x-x_0)^n", + "\\end{equation}", + "in some interval containing $x_0$. Since $\\lim_{x\\to x_0} E_n(x)=0$", + "and", + "$f^{(n)}(x_0)\\ne0$, there is a $\\delta>0$ such that", + "$$", + "|E_n(x)|<\\left|\\frac{f^{(n)}(x_0)}{ n!}\\right|\\mbox{\\quad if\\quad}", + "|x-x_0|", + "<\\delta.", + "$$", + "\\newpage", + "\\noindent", + "This and \\eqref{eq:2.5.10} imply that", + "\\begin{equation}\\label{eq:2.5.11}", + "\\frac{f(x)-f(x_0)}{(x-x_0)^n}", + "\\end{equation}", + "has the same sign as $f^{(n)}(x_0)$ if $0<|x-x_0|<\\delta$. If $n$ is", + "odd the denominator of \\eqref{eq:2.5.11} changes sign in every", + "neighborhood of $x_0$, and therefore so must the numerator (since the", + "ratio has constant sign for $0<|x-x_0|<\\delta$). Consequently,", + "$f(x_0)$ cannot be a local extreme value of $f$. This proves \\part{a}. If", + "$n$ is even, the denominator of \\eqref{eq:2.5.11} is positive for $x\\ne", + "x_0$, so $f(x)-f(x_0)$ must have the same sign as", + "$f^{(n)}(x_0)$ for $0<|x-x_0|<\\delta$. This proves \\part{b}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 42, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", + "categories": [], + "title": "Taylor's Theorem", + "contents": [ + "Suppose that $f^{(n+1)}$ exists on an open interval $I$ about $x_0,$", + "and let", + "$x$ be in $I.$ Then the remainder", + "$$", + "R_n(x)=f(x)-T_n(x)", + "$$", + "can be written as", + "$$", + "R_n(x)=\\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1},", + "$$", + "where $c$ depends upon $x$ and is between $x$ and $x_0.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 43, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.5", + "categories": [], + "title": "Extended Mean Value Theorem", + "contents": [ + "Suppose that $f$ is continuous on a finite closed interval $I$ with", + "endpoints $a$ and $b$ $($that is, either $I=(a,b)$ or $I=(b,a)),$", + "$f^{(n+1)}$ exists on the open interval $I^0,$ and$,$ if $n>0,$ that", + "$f'$, \\dots, $f^{(n)}$ exist and are continuous at $a.$ Then", + "\\begin{equation}\\label{eq:2.5.17}", + "f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=\\frac{f^{(n+1)}(c)}{(n+1)!}", + "(b-a)^{n+1}", + "\\end{equation}", + "for some $c$ in $I^0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The proof is by induction. The mean value theorem", + "(Theorem~\\ref{thmtype:2.3.11}) implies the conclusion for $n=0$.", + "Now suppose that", + "$n\\ge1$, and assume that the assertion of the theorem is true with $n$", + "replaced by", + "$n-1$. The left side of \\eqref{eq:2.5.17} can be written as", + "\\begin{equation}\\label{eq:2.5.18}", + "f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=K\\frac{(b-a)^{n+1}}{(n+1)!}", + "\\end{equation}", + "for some number $K$. We must prove that $K=f^{(n+1)}(c)$ for", + "some $c$ in $I^0$. To this end, consider the auxiliary function", + "$$", + "h(x)=f(x)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{", + "r!}(x-a)^r-K\\frac{(x-a)^{n+1}}{", + "(n+1)!},", + "$$", + "which satisfies", + "$$", + "h(a)=0,\\quad h(b)=0,", + "$$", + "(the latter because of \\eqref{eq:2.5.18}) and is continuous on the closed", + "interval $I$ and differentiable on $I^0$, with", + "\\begin{equation}\\label{eq:2.5.19}", + "h'(x)=f'(x)-\\sum_{r=0}^{n-1}\\frac{f^{(r+1)}(a)}{", + "r!}(x-a)^r-K\\frac{(x-a)^n}{n!}.", + "\\end{equation}", + "Therefore, Rolle's theorem (Theorem~\\ref{thmtype:2.3.8})", + "implies that $h'(b_1)=0$ for some $b_1$ in", + "$I^0$; thus, from \\eqref{eq:2.5.19},", + "$$", + "f'(b_1)-\\sum_{r=0}^{n-1}\\frac{f^{(r+1)}(a)}{", + "r!}(b_1-a)^r-K\\frac{(b_1-a)^n}{n!}=0.", + "$$", + "If we temporarily write $f'=g$, this becomes", + "\\begin{equation}\\label{eq:2.5.20}", + "g(b_1)-\\sum_{r=0}^{n-1}\\frac{g^{(r)}(a)}{", + "r~}(b_1-a)^r-K\\frac{(b_1-a)^n}{n!}=0.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Since $b_1\\in I^0$, the hypotheses on $f$ imply that $g$ is continuous", + "on the closed interval $J$ with endpoints $a$ and $b_1$, $g^{(n)}$", + "exists on", + "$J^0$, and, if $n\\ge1$, $g'$, \\dots, $g^{(n-1)}$ exist and are", + "continuous", + "at $a$ (also at $b_1$, but this is not important). The induction", + "hypothesis, applied to $g$ on the interval $J$, implies that", + "$$", + "g(b_1)-\\sum_{r=0}^{n-1}\\frac{g^{(r)}(a)}{ r!}", + "(b_1-a)^r=\\frac{g^{(n)}(c)}{n!}(b_1-a)^n", + "$$", + "for some $c$ in $J^0$. Comparing this with \\eqref{eq:2.5.20} and recalling", + "that $g=f'$ yields", + "$$", + "K=g^{(n)}(c)=f^{(n+1)}(c).", + "$$", + "Since $c$ is in $I^0$, this completes the induction." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.3.11", + "TRENCH_REAL_ANALYSIS-thmtype:2.3.8" + ], + "ref_ids": [ + 35, + 32 + ] + } + ], + "ref_ids": [] + }, + { + "id": 44, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.2", + "categories": [], + "title": "", + "contents": [ + "If $f$ is unbounded on $[a,b],$ then $f$ is not integrable on", + "$[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will show that if $f$ is unbounded on $[a,b]$, $P$ is any", + "partition of $[a,b]$, and $M>0$, then there are Riemann sums $\\sigma$", + "and $\\sigma'$ of $f$ over $P$ such that", + "\\begin{equation} \\label{eq:3.1.7}", + "|\\sigma-\\sigma'|\\ge M.", + "\\end{equation}", + "We leave it to you (Exercise~\\ref{exer:3.1.2}) to complete the proof by", + "showing from this that", + "$f$ cannot satisfy Definition~\\ref{thmtype:3.1.1}.", + "Let", + "$$", + "\\sigma=\\sum_{j=1}^nf(c_j)(x_j-x_{j-1})", + "$$", + "be a Riemann sum of $f$ over a partition $P$ of $[a,b]$. There must be", + "an integer $i$ in $\\{1,2, \\dots,n\\}$ such that", + "\\begin{equation} \\label{eq:3.1.8}", + "|f(c)-f(c_i)|\\ge \\frac{M }{ x_i-x_{i-1}}", + "\\end{equation}", + "for some $c$ in $[x_{i-1}x_i]$, because if there were not so, we", + "would have", + "$$", + "|f(x)-f(c_j)|<\\frac{M}{ x_j-x_{j-1}},\\quad x_{j-1}\\le x\\le x_j,\\quad", + "1\\le j\\le n.", + "$$", + "Then", + "\\begin{eqnarray*}", + "|f(x)|\\ar=|f(c_j)+f(x)-f(c_j)|\\le|f(c_j)|+|f(x)-f(c_j)|\\\\", + "\\ar\\le |f(c_j)|+\\frac{M}{ x_j-x_{j-1}},\\quad x_{j-1}\\le x\\le x_j,\\quad", + "1\\le j\\le n.", + "\\end{eqnarray*}", + "which implies that", + "$$", + "|f(x)|\\le\\max_{1\\le j\\le n}|f(c_j)|+\\frac{M}{", + "x_j-x_{j-1}},", + "\\quad a\\le x \\le b,", + "$$", + "contradicting the assumption that $f$ is unbounded on $[a,b]$.", + " Now suppose that $c$ satisfies \\eqref{eq:3.1.8}, and", + "consider the Riemann sum", + "$$", + "\\sigma'=\\sum_{j=1}^nf(c'_j)(x_j-x_{j-1})", + "$$", + "over the same partition $P$, where", + "$$", + "c'_j=\\left\\{\\casespace\\begin{array}{ll}", + "c_j,&j \\ne i,\\\\", + "c,&j=i.\\end{array}\\right.", + "$$", + "\\newpage", + "\\noindent", + "Since", + "$$", + "|\\sigma-\\sigma'|=|f(c)-f(c_i)|(x_i-x_{i-1}),", + "$$", + "\\eqref{eq:3.1.8} implies \\eqref{eq:3.1.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.1.1" + ], + "ref_ids": [ + 315 + ] + } + ], + "ref_ids": [] + }, + { + "id": 45, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.4", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be bounded on $[a,b]$, and let $P$", + "be a partition of $[a,b].$ Then", + "\\begin{alist}", + "\\item % (a)", + " The upper sum $S(P)$ of $f$ over $P$ is the supremum", + " of the set of all Riemann sums of $f$ over $P.$", + "\\item % (b)", + " The lower sum $s(P)$ of $f$ over $P$ is the infimum", + " of the set of all Riemann sums of $f$ over $P.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} If $P=\\{x_0,x_1, \\dots,x_n\\}$, then", + "$$", + "S(P)=\\sum_{j=1}^n M_j(x_j-x_{j-1}),", + "$$", + "where", + "$$", + "M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x).", + "$$", + "An arbitrary Riemann sum of $f$ over $P$ is of the form", + "$$", + "\\sigma=\\sum_{j=1}^n f(c_j)(x_j-x_{j-1}),", + "$$", + "where $x_{j-1}\\le c_j\\le x_j$.", + "Since $f(c_j)\\le M_j$, it follows that $\\sigma\\le S(P)$.", + "Now let", + "$\\epsilon>0$ and choose $\\overline c_j$ in $[x_{j-1},x_j]$ so that", + "$$", + "f(\\overline c_j) > M_j -\\frac{\\epsilon}{ n(x_j-x_{j-1})},\\quad 1\\le j\\le", + "n.", + "$$", + "The Riemann sum produced in this way is", + "$$", + "\\overline \\sigma=\\sum_{j=1}^n", + "f(\\overline", + "c_j)(x_j-x_{j-1})>\\sum_{j=1}^n\\left[M_j-\\frac{\\epsilon}{", + "n(x_j-x_{j-1})})\\right](x_j-x_{j-1})=S(P)-\\epsilon.", + "$$", + "Now Theorem~\\ref{thmtype:1.1.3} implies that", + "$S(P)$ is the supremum of the set of Riemann sums of $f$", + "over $P$.", + "\\part{b} Exercise~\\ref{exer:3.1.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.1.3" + ], + "ref_ids": [ + 1 + ] + } + ], + "ref_ids": [] + }, + { + "id": 46, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.2", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on $[a,b],$ then", + "\\begin{equation} \\label{eq:3.2.6}", + "\\underline{\\int_a^b}f(x)\\,dx\\le\\overline{\\int_a^b}f(x)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $P_1$ and $P_2$ are partitions of $[a,b]$ and $P'$ is a", + "refinement of both. Letting $P=P_1$ in \\eqref{eq:3.2.3} and $P=P_2$ in", + "\\eqref{eq:3.2.2} shows that", + "$$", + "s(P_1)\\le s(P') \\mbox{\\quad and \\quad} S(P')\\le S(P_2).", + "$$", + "Since $s(P')\\le S(P')$, this implies that", + "$s(P_1)\\le S(P_2)$.", + "Thus, every lower sum is a lower bound for the set of all upper sums.", + "Since $\\overline{\\int_a^b}f(x)\\,dx$ is the infimum of", + "this set, it follows that", + "$$", + "s(P_1)\\le\\overline{\\int_a^b}f(x)\\,dx", + "$$", + "for every partition $P_1$ of $[a,b]$. This means that", + "$\\overline{\\int_a^b}", + "f(x)\\,dx$ is an upper bound for the set of all lower sums. Since", + "$\\underline{\\int_a^b} f(x)\\,dx$ is the supremum of this set,", + "this implies \\eqref{eq:3.2.6}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 47, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.3", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b],$ then", + "$$", + "\\underline{\\int_a^b}f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b", + "f(x)\\,dx.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We prove that", + "$\\overline{\\int_a^b}f(x)\\,dx=\\int_a^bf(x)\\,dx$ and leave it to you to", + "show that", + "$\\underline{\\int_a^b}f(x)\\,dx=\\int_a^bf(x)\\,dx$", + "(Exercise~\\ref{exer:3.2.2}).", + " Suppose that $P$ is a partition of $[a,b]$", + "and $\\sigma$ is a Riemann sum of $f$ over $P$.", + "Since", + "\\begin{eqnarray*}", + "\\overline{\\int_a^b}f(x)\\,dx-\\int_a^b f(x)\\,dx\\ar=", + "\\left(\\overline{\\int_a^b}f(x)\\,dx-S(P)\\right)+(S(P)-\\sigma)", + "\\\\[2\\jot]", + "&&+\\left(\\sigma-\\int_a^b f(x)\\ dx\\right),", + "\\end{eqnarray*}", + "\\newpage", + "\\noindent", + "the triangle inequality implies that", + "\\begin{equation} \\label{eq:3.2.7}", + "\\begin{array}{rcl}", + "\\dst{\\left|\\overline{\\int_a^b}f(x)\\,dx-\\int_a^b f(x)\\,dx \\right|}\\ar\\le", + "\\dst{\\left|\\overline{\\int_a^b}f(x)\\,dx-S(P)\\right|+|S(P)-\\sigma|}", + "\\\\[2\\jot]", + "&&+\\dst{\\left|\\sigma-\\int_a^b f(x)\\ dx\\right|}.", + "\\end{array}", + "\\end{equation}", + "Now suppose that $\\epsilon>0$.", + " From Definition~\\ref{thmtype:3.1.3}, there is", + "a partition $P_0$ of $[a,b]$ such that", + "\\begin{equation} \\label{eq:3.2.8}", + "\\overline{\\int_a^b} f(x)\\,dx\\le S(P_0)<", + "\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{3}.", + "\\end{equation}", + "From Definition~\\ref{thmtype:3.1.1}, there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:3.2.9}", + "\\left|\\sigma-\\int_a^bf(x)\\,dx\\right|<\\frac{\\epsilon}{3}", + "\\end{equation}", + "if $\\|P\\|<\\delta$. Now suppose that $\\|P\\|<\\delta$ and $P$ is a", + "refinement of $P_0$. Since $S(P)\\le S(P_0)$ by Lemma~\\ref{thmtype:3.2.1},", + "\\eqref{eq:3.2.8} implies that", + "$$", + "\\overline{\\int_a^b} f(x)\\,dx\\le S(P)<", + "\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{3},", + "$$", + "so", + "\\begin{equation} \\label{eq:3.2.10}", + "\\left|S(P)-\\overline{\\int_a^b}f(x)\\,dx\\right|<\\frac{\\epsilon}{3}", + "\\end{equation}", + "in addition to \\eqref{eq:3.2.9}. Now \\eqref{eq:3.2.7}, \\eqref{eq:3.2.9}, and", + "\\eqref{eq:3.2.10} imply that", + "\\begin{equation} \\label{eq:3.2.11}", + "\\left|\\overline{\\int_a^b} f(x)\\,dx-\\int_a^b f(x)\\,dx\\right|<", + "\\frac{2\\epsilon}{3}+|S(P)-\\sigma|", + "\\end{equation}", + "for every Riemann sum $\\sigma$ of $f$ over $P$. Since $S(P)$ is the", + "supremum of these Riemann sums", + "(Theorem~\\ref{thmtype:3.1.4}), we may choose", + "$\\sigma$ so that", + "$$", + "|S(P)-\\sigma|<\\frac{\\epsilon}{3}.", + "$$", + "Now \\eqref{eq:3.2.11} implies that", + "$$", + "\\left|\\overline{\\int_a^b} f(x)\\,dx-\\int_a^b f(x)\\,dx \\right|<", + "\\epsilon.", + "$$", + "Since $\\epsilon$ is an arbitrary positive number, it follows that", + "$$", + "\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b f(x)\\,dx.", + "$$", + "\\vskip-6.5ex" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.1.4" + ], + "ref_ids": [ + 316, + 315, + 246, + 45 + ] + } + ], + "ref_ids": [] + }, + { + "id": 48, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.5", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on $[a,b]$ and", + "\\begin{equation} \\label{eq:3.2.16}", + "\\underline{\\int_a^b} f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=L,", + "\\end{equation}", + "then $f$ is integrable on $[a,b]$ and", + "\\begin{equation} \\label{eq:3.2.17}", + "\\int_a^b f(x)\\,dx=L.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $\\epsilon>0$, there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:3.2.18}", + "\\underline{\\int_a^b}f(x)\\,dx-\\epsilon0$ there is", + "a partition $P$ of $[a,b]$ for which", + "\\begin{equation} \\label{eq:3.2.19}", + "S(P)-s(P)<\\epsilon.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We leave it to you (Exercise~\\ref{exer:3.2.4}) to show that if $\\int_a^b", + "f(x)\\,dx$ exists, then \\eqref{eq:3.2.19} holds for $\\|P\\|$ sufficiently", + "small. This implies that the stated condition is necessary for", + "integrability. To show that it is sufficient, we observe that since", + "$$", + "s(P)\\le \\underline{\\int_a^b}f(x)\\,dx\\le\\overline{\\int_a^b}f(x)\\,dx\\le", + "S(P)", + "$$", + "for all $P$, \\eqref{eq:3.2.19} implies that", + "$$", + "0\\le\\overline{\\int_a^b} f(x)\\,dx-\\underline{\\int_a^b}f(x)\\,dx<", + "\\epsilon.", + "$$", + "Since $\\epsilon$ can be any positive number, this implies that", + "$$", + "\\overline{\\int_a^b} f(x)\\,dx=\\underline{\\int_a^b} f(x)\\,dx.", + "$$", + "Therefore, $\\int_a^b f(x)\\,dx$ exists, by Theorem~\\ref{thmtype:3.2.5}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.2.5" + ], + "ref_ids": [ + 48 + ] + } + ], + "ref_ids": [] + }, + { + "id": 51, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on $[a,b],$", + "then $f$ is integrable on $[a,b]$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $P=\\{x_0,x_1, \\dots,x_n\\}$ be a partition of $[a,b]$. Since", + "$f$ is continuous on $[a,b]$, there are points $c_j$ and $c'_j$ in", + "$[x_{j-1},x_j]$ such that", + "$$ f(c_j)=M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x)", + "$$", + "and", + "$$", + "f(c'_j)=m_j=\\inf_{x_{j-1}\\le x\\le x_j}f(x)", + "$$", + "(Theorem~\\ref{thmtype:2.2.9}).", + "Therefore,", + "\\begin{equation} \\label{eq:3.2.20}", + "S(P)-s(P)=\\sum_{j=1}^n\\left[f(c_j)-f(c'_j)\\right](x_j-x_{j-1}).", + "\\end{equation}", + "Since $f$ is uniformly continuous on $[a,b]$", + "(Theorem~\\ref{thmtype:2.2.12}), there is for each $\\epsilon>0$", + "a", + "$\\delta>0$ such that", + " $$", + "|f(x')-f(x)|<\\frac{\\epsilon}{ b-a}", + " $$", + " if $x$ and $x'$ are", + "in $[a,b]$ and $|x-x'|<\\delta$. If $\\|P\\|<\\delta$, then", + "$|c_j-c'_j|<\\delta$ and, from \\eqref{eq:3.2.20},", + "$$", + " S(P)-s(P)<\\frac{\\epsilon}{ b-a}", + "\\sum_{j=1}^n(x_j-x_{j-1})=\\epsilon.", + "$$", + "Hence, $f$ is integrable", + "on $[a,b]$, by Theorem~\\ref{thmtype:3.2.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", + "TRENCH_REAL_ANALYSIS-thmtype:2.2.12", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" + ], + "ref_ids": [ + 23, + 25, + 50 + ] + } + ], + "ref_ids": [] + }, + { + "id": 52, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.9", + "categories": [], + "title": "", + "contents": [ + "If $f$ is monotonic on $[a,b],$ then $f$ is integrable on $[a,b]$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $P=\\{x_0,x_1, \\dots,x_n\\}$ be a partition of $[a,b]$. Since", + " $f$ is nondecreasing,", + "\\begin{eqnarray*}", + "f(x_j)\\ar=M_j=\\sup_{x_{j-1}\\le x\\le x_j}f(x)\\\\", + "\\arraytext{and}\\\\", + "f(x_{j-1})\\ar=m_j=\\inf_{x_{j-1}\\le x\\le x_j}f(x).", + "\\end{eqnarray*}", + "Hence,", + "$$", + "S(P)-s(P)=\\sum_{j=1}^n(f(x_j)-f(x_{j-1})) (x_j-x_{j-1}).", + "$$", + "Since $00$", + "there are positive numbers $\\delta_1$ and $\\delta_2$ such that", + "\\begin{eqnarray*}", + "\\left|\\sigma_f-\\int_a^b f(x)\\,dx\\right|\\ar<\\frac{\\epsilon}{2}", + "\\mbox{\\quad if\\quad}\\|P\\|<\\delta_1\\\\", + "\\arraytext{and}\\\\", + "\\left|\\sigma_g-\\int_a^b g(x)\\,dx\\right|\\ar<\\frac{\\epsilon}{2}", + "\\mbox{\\quad if\\quad}\\|P\\|<\\delta_2.", + "\\end{eqnarray*}", + "If $\\|P\\|<\\delta=\\min(\\delta_1,\\delta_2)$, then", + "\\begin{eqnarray*}", + "\\left|\\sigma_{f+g}-\\int_a^b f(x)\\,dx-\\int_a^b g(x)\\,dx\\right|", + "\\ar=\\left|\\left(\\sigma_f-\\int_a^b f(x)\\,dx\\right)+", + "\\left(\\sigma_g-\\int_a^b g(x)\\,dx\\right)\\right|\\\\", + "\\ar\\le \\left|\\sigma_f-\\int_a^b f(x)\\,dx\\right|+", + "\\left|\\sigma_g-\\int_a^b g(x)\\,dx\\right|\\\\", + "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon,", + "\\end{eqnarray*}", + "so the conclusion follows from Definition~\\ref{thmtype:3.1.1}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.1.1" + ], + "ref_ids": [ + 315, + 315 + ] + } + ], + "ref_ids": [] + }, + { + "id": 54, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b]$ and", + "$c$ is a constant$,$ then $cf$ is integrable on $[a,b]$ and", + "$$", + "\\int_a^b cf(x)\\,dx=c\\int_a^b f(x)\\,dx.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 55, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.3", + "categories": [], + "title": "", + "contents": [ + " If $f_1,$ $f_2,$ \\dots$,$ $f_n$ are", + "integrable on $[a,b]$ and $c_1,$ $c_2,$ \\dots$,$ $c_n$ are", + "constants$,$ then", + "$c_1f_1+c_2f_2+\\cdots+ c_nf_n$ is integrable on $[a,b]$ and", + "\\begin{eqnarray*}", + "\\int_a^b (c_1f_1+c_2f_2+\\cdots+c_nf_n)(x)\\,dx\\ar=c_1\\int_a^b f_1(x)\\,dx", + "+c_2\\int_a^b f_2(x)\\,dx\\\\", + "\\ar{}+\\cdots+c_n\\int_a^b f_n(x)\\,dx.", + "\\end{eqnarray*}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 56, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are integrable on", + "$[a,b]$ and $f(x)\\le g(x)$ for $a\\le x\\le b,$ then", + "\\begin{equation}\\label{eq:3.3.1}", + "\\int_a^b f(x)\\,dx\\le\\int_a^b g(x)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $g(x)-f(x)\\ge0$, every lower sum of $g-f$ over any", + "partition of $[a,b]$ is nonnegative. Therefore,", + "$$", + "\\underline{\\int_a^b}(g(x)-f(x))\\,dx\\ge0.", + "$$", + "Hence,", + "\\begin{equation}\\label{eq:3.3.2}", + "\\begin{array}{rcl}", + "\\dst\\int_a^b g(x)\\,dx-\\int_a^b f(x)\\,dx\\ar=\\dst\\int_a^b", + "(g(x)-f(x))\\,dx\\\\[2\\jot]", + "\\ar=\\dst\\underline{\\int_a^b}(g(x)-f(x))\\,dx\\ge0,", + "\\end{array}", + "\\end{equation}", + "which yields \\eqref{eq:3.3.1}. (The first equality in \\eqref{eq:3.3.2}", + "follows", + "from Theorems~\\ref{thmtype:3.3.1} and \\ref{thmtype:3.3.2}; the second, from", + "Theorem~\\ref{thmtype:3.2.3}.)" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.3.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.3" + ], + "ref_ids": [ + 53, + 54, + 47 + ] + } + ], + "ref_ids": [] + }, + { + "id": 57, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.5", + "categories": [], + "title": "", + "contents": [ + " If $f$ is integrable on $[a,b],$", + "then so is $|f|$, and", + "\\begin{equation} \\label{eq:3.3.3}", + "\\left|\\int_a^b f(x)\\,dx\\right|\\le\\int_a^b |f(x)|\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $P$ be a partition of $[a,b]$ and define", + "\\begin{eqnarray*}", + "M_j\\ar=\\sup\\set{f(x)}{x_{j-1}\\le x\\le x_j},\\\\", + "m_j\\ar=", + "\\inf\\set{f(x)}{x_{j-1}\\le x\\le x_j},\\\\", + "\\overline{M}_j\\ar=\\sup\\set{|f(x)|}{x_{j-1}\\le x\\le x_j},\\\\", + "\\overline{m}_j\\ar=\\inf\\set{|f(x)|}{x_{j-1}\\le x\\le x_j}.", + "\\end{eqnarray*}", + "Then", + "\\begin{equation} \\label{eq:3.3.4}", + "\\begin{array}{rcl}", + "\\overline{M}_j-\\overline{m}_j\\ar=", + "\\dst\\sup\\set{|f(x)|-|f(x')|}{x_{j-1}\\le x,x'\\le x_j}\\\\", + "\\ar\\le \\dst\\sup\\set{|f(x)-f(x')|}{x_{j-1}\\le x,x'\\le x_j}\\\\", + "\\ar=M_j-m_j.", + "\\end{array}", + "\\end{equation}", + "Therefore,", + "$$", + "\\overline{S}(P)-\\overline{s}(P)\\le S(P)-s(P),", + "$$", + "where the upper and lower sums on the left are associated with $|f|$", + "and those on the right are associated with $f$. Now suppose that", + "$\\epsilon>0$. Since $f$ is integrable on $[a,b]$,", + " Theorem~\\ref{thmtype:3.2.7} implies that", + "there is a partition $P$ of $[a,b]$ such that $S(P)-s(P)<\\epsilon$.", + "This inequality and \\eqref{eq:3.3.4} imply that $\\overline", + "S(P)-\\overline s(P)<\\epsilon$.", + " Therefore, $|f|$ is integrable on $[a,b]$,", + " again by Theorem~\\ref{thmtype:3.2.7}.", + "Since", + "$$", + "f(x)\\le|f(x)|\\mbox{\\quad and \\quad}-f(x)\\le|f(x)|,\\quad a\\le x\\le b,", + "$$", + "\\newpage", + "\\noindent", + " Theorems~\\ref{thmtype:3.3.2} and \\ref{thmtype:3.3.4} imply", + "that", + "$$", + "\\int_a^b f(x)\\,dx\\le\\int_a^b|f(x)|\\,dx\\mbox{\\quad and }", + "-\\int_a^b f(x)\\,dx\\le\\int_a^b|f(x)|\\,dx,", + "$$", + "which implies \\eqref{eq:3.3.3}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.4" + ], + "ref_ids": [ + 50, + 50, + 54, + 56 + ] + } + ], + "ref_ids": [] + }, + { + "id": 58, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are integrable on $[a,b],$ then so is the product", + "$fg.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We consider the case where $f$ and $g$ are nonnegative, and", + "leave the rest of the proof to you (Exercise~\\ref{exer:3.3.4}). The", + "subscripts $f$, $g$, and $fg$ in the following argument identify the", + "functions", + "with which the various quantities are associated. We assume that", + "neither $f$ nor $g$ is identically zero on $[a,b]$, since the", + "conclusion is obvious if one of them is.", + "If $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, then", + "\\begin{equation}\\label{eq:3.3.5}", + "S_{fg}(P)-s_{fg}(p)=\\sum_{j=1}^n (M_{fg,j}-m_{fg,", + "j})(x_j-x_{j-1}).", + "\\end{equation}", + "Since $f$ and $g$ are nonnegative, $M_{fg,j}\\le M_{f,j}M_{g,j}$ and", + "$m_{fg,j}\\ge m_{f,j}m_{g,j}$. Hence,", + "\\begin{eqnarray*}", + "M_{fg,j}-m_{fg,j}\\ar\\le M_{f,j}M_{g,j}-m_{f,", + "j}m_{g,j}\\\\[2\\jot]", + "\\ar=(M_{f,j}-m_{f,j})M_{g,j}+m_{f,j}(M_{g,j}-", + "m_{g,j})\\\\[2\\jot]", + "\\ar\\le M_g(M_{f,j}-m_{f,j})+M_f(M_{g,j}-m_{g,j}),", + "\\end{eqnarray*}", + "where $M_f$ and $M_g$ are upper bounds for $f$ and $g$ on $[a,b]$. From", + "\\eqref{eq:3.3.5} and the last inequality,", + "\\begin{equation} \\label{eq:3.3.6}", + "S_{fg}(P)-s_{fg}(P)\\le M_g[S_f(P)-s_f(P)]+M_f[S_g(P)-s_g(P)].", + "\\end{equation}", + "Now suppose that $\\epsilon>0$. Theorem~\\ref{thmtype:3.2.7}", + "implies that there are partitions $P_1$ and $P_2$ of $[a,b]$ such that", + "\\begin{equation} \\label{eq:3.3.7}", + "S_f(P_1)-s_f(P_1)<\\frac{\\epsilon}{2M_g}\\mbox{\\quad and\\quad}", + "S_g(P_2)-s_g(P_2)<\\frac{\\epsilon}{2M_f}.", + "\\end{equation}", + "If $P$ is a refinement of both $P_1$ and $P_2$,", + " then \\eqref{eq:3.3.7}", + "and Lemma~\\ref{thmtype:3.2.1} imply that", + "$$", + "S_f(P)-s_f(P)<\\frac{\\epsilon}{2M_g}\\mbox{\\quad and\\quad}", + "S_g(P)-s_g(P)<\\frac{\\epsilon}{2M_f}.", + "$$", + "This and \\eqref{eq:3.3.6} yield", + "$$", + "S_{fg}(P)-s_{fg}(P)<\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon.", + "$$", + " Therefore, $fg$ is integrable on $[a,b]$, by", + "Theorem~\\ref{thmtype:3.2.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" + ], + "ref_ids": [ + 50, + 246, + 50 + ] + } + ], + "ref_ids": [] + }, + { + "id": 59, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.7", + "categories": [], + "title": "First Mean Value Theorem for Integrals", + "contents": [ + "Suppose that $u$ is continuous and $v$ is integrable and nonnegative", + "on", + "$[a,b].$ Then", + "\\begin{equation} \\label{eq:3.3.8}", + "\\int_a^b u(x)v(x)\\,dx=u(c)\\int_a^b v(x)\\,dx", + "\\end{equation}", + "for some $c$ in $[a,b]$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Theorem~\\ref{thmtype:3.2.8}, $u$ is integrable on", + "$[a,b]$. Therefore,", + "Theorem~\\ref{thmtype:3.3.6} implies", + "that the integral on the left exists. If $m=\\min\\set{u(x)}{a\\le x\\le", + "b}$", + " and $M=\\max\\set{u(x)}{a\\le x\\le b}$ (recall", + "Theorem~\\ref{thmtype:2.2.9}), then", + "$$", + "m\\le u(x)\\le M", + "$$", + "and, since $v(x)\\ge0$,", + "$$", + "mv(x)\\le u(x) v(x)\\le Mv(x).", + "$$", + "Therefore, Theorems~\\ref{thmtype:3.3.2} and", + "\\ref{thmtype:3.3.4} imply that", + "\\vskip2pt", + "\\begin{equation} \\label{eq:3.3.9}", + "m\\int_a^b v(x)\\,dx\\le\\int_a^b u(x)v(x)\\,dx\\le M\\int_a^b v(x)\\,dx.", + "\\end{equation}", + "\\vskip2pt", + "This implies that \\eqref{eq:3.3.8} holds for any $c$ in $[a,b]$", + "if $\\int_a^b v(x)\\,dx=0$. If $\\int_a^b v(x)\\,dx\\ne0$, let", + "\\vskip1pt", + "\\begin{equation} \\label{eq:3.3.10}", + "\\overline{u}=\\frac{\\dst\\int_a^b u(x)v(x)\\,dx}{\\dst\\int_a^bv(x)\\,dx}", + "\\end{equation}", + "\\vskip1pt", + "\\noindent Since $\\int_a^b v(x)\\,dx>0$ in this case (why?),", + "\\eqref{eq:3.3.9} implies", + "that $m\\le\\overline{u}\\le M$, and the intermediate value theorem", + " (Theorem~\\ref{thmtype:2.2.10}) implies that $\\overline{u}=u(c)$", + "for some $c$ in $[a,b]$. This implies \\eqref{eq:3.3.8}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.2.8", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", + "TRENCH_REAL_ANALYSIS-thmtype:2.2.9", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.2", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", + "TRENCH_REAL_ANALYSIS-thmtype:2.2.10" + ], + "ref_ids": [ + 51, + 58, + 23, + 54, + 56, + 24 + ] + } + ], + "ref_ids": [] + }, + { + "id": 60, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b]$", + "and $a\\le a_10$. From Theorem~\\ref{thmtype:3.2.7},", + "there is a partition $P=\\{x_0,x_1, \\dots,x_n\\}$ of $[a,b]$ such that", + "\\begin{equation} \\label{eq:3.3.11}", + "S(P)-s(P)=\\sum_{j=1}^n(M_j-m_j)(x_j-x_{j-1})<\\epsilon.", + "\\end{equation}", + "We may assume that $a_1$ and $b_1$ are partition points of $P$,", + "because if not they can be inserted to obtain a refinement", + "$P'$ such that $S(P')-s(P')\\le S(P)-s(P)$", + "(Lemma~\\ref{thmtype:3.2.1}). Let", + "$a_1=x_r$ and $b_1=x_s$. Since every term in \\eqref{eq:3.3.11} is", + "nonnegative,", + "$$", + "\\sum_{j=r+1}^s (M_j-m_j)(x_j-x_{j-1})<\\epsilon.", + "$$", + "Thus, $\\overline{P}=\\{x_r,x_{r+1}, \\dots,x_s\\}$ is a partition of", + "$[a_1,b_1]$ over which the upper and lower sums of $f$ satisfy", + "$$", + "S(\\overline{P})-s(\\overline{P})<\\epsilon.", + "$$", + " Therefore, $f$ is integrable on $[a_1,b_1]$, by", + "Theorem~\\ref{thmtype:3.2.7}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" + ], + "ref_ids": [ + 50, + 246, + 50 + ] + } + ], + "ref_ids": [] + }, + { + "id": 61, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.9", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b]$", + "and $[b,c],$ then $f$ is integrable on $[a,c],$ and", + "\\begin{equation} \\label{eq:3.3.12}", + "\\int_a^cf(x)\\,dx=\\int_a^bf(x)\\,dx+\\int_b^cf(x)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 62, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.10", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b]$ and", + "$a\\le c\\le b,$ then the function", + "$F$ defined by", + "$$", + " F(x)=\\int_c^x f(t)\\,dt", + "$$", + " satisfies a Lipschitz", + "condition on $[a,b],$ and is therefore", + "continuous on", + "$[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $x$ and $x'$ are in $[a,b]$, then", + "$$", + "F(x)-F(x')=\\int_c^x f(t)\\,dt-\\int_c^{x'} f(t)\\,dt=\\int_{x'}^x f(t)\\,", + "dt,", + "$$", + "by Theorem~\\ref{thmtype:3.3.9} and the conventions just adopted. Since", + "$|f(t)|\\le K$ $(a\\le t\\le b)$ for some constant $K$,", + "$$", + "\\left|\\int_{x'}^x f(t)\\,dt\\right|\\le K|x-x'|,\\quad a\\le x,\\, x'\\le b", + "$$", + "(Theorem~\\ref{thmtype:3.3.5}), so", + "$$", + "|F(x)-F(x')|\\le K|x-x'|,\\quad a\\le x,\\,x'\\le b.", + "$$", + "\\vskip-2em" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.3.9", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.5" + ], + "ref_ids": [ + 61, + 57 + ] + } + ], + "ref_ids": [] + }, + { + "id": 63, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.11", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then", + "$F(x)=\\int_c^x", + "f(t)\\,dt$ is differentiable at any point $x_0$ in $(a,b)$ where $f$ is", + "continuous$,$ with $F'(x_0)=f(x_0).$ If $f$ is continuous from the", + "right at $a,$ then $F_+'(a)=f(a)$. If $f$ is continuous from", + "the left at $b,$ then $F_-'(b)=f(b).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We consider the case where $a0$ a", + "$\\delta>0$ such that", + "$$", + "|f(t)-f(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|<\\delta", + "$$", + "and $t$ is between $x$ and $x_0$. Therefore, from \\eqref{eq:3.3.13},", + "$$", + "\\left|\\frac{F(x)-F(x_0)}{ x-x_0}-f(x_0)\\right|<\\epsilon", + "\\frac{|x-x_0|}{", + "|x-x_0|}=\\epsilon\\mbox{\\quad if\\quad} 0<|x-x_0|<\\delta.", + "$$", + "Hence, $F'(x_0)=f(x_0)$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.3.5" + ], + "ref_ids": [ + 57 + ] + } + ], + "ref_ids": [] + }, + { + "id": 64, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.12", + "categories": [], + "title": "", + "contents": [ + "Suppose that $F$ is continuous on the closed interval $[a,b]$ and", + "differentiable on the open interval", + "$(a,b),$ and $f$ is integrable on $[a,b].$ Suppose also that", + "$$", + "F'(x)=f(x),\\quad a0$ a $\\delta>0$", + "such that", + "$$", + "\\left|\\sigma-\\int_a^b f(x)\\,dx\\right|<\\epsilon\\mbox{\\quad if\\quad}", + "\\|P\\|<\\delta.", + "$$", + "Therefore,", + "$$", + "\\left|F(b)-F(a)-\\int_a^b f(x)\\,dx\\right|<\\epsilon", + "$$", + "for every $\\epsilon>0$, which implies \\eqref{eq:3.3.14}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.3.11" + ], + "ref_ids": [ + 35 + ] + } + ], + "ref_ids": [] + }, + { + "id": 65, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.14", + "categories": [], + "title": "Fundamental Theorem of Calculus", + "contents": [ + "If $f$ is continuous on $[a,b],$ then $f$ has an antiderivative on", + "$[a,b].$ Moreover$,$ if $F$ is any antiderivative of $f$ on $[a,b],$", + "then", + "$$", + "\\int_a^b f(x)\\,dx=F(b)-F(a).", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The function", + " $F_0(x)=\\int_a^x f(t)\\,dt$ is", + "continuous on $[a,b]$ by Theorem~\\ref{thmtype:3.3.10}, and $F_0'(x)", + "=f(x)$ on $(a,b)$ by Theorem~\\ref{thmtype:3.3.11}. Therefore,", + "$F_0$ is an antiderivative of $f$ on $[a,b]$.", + "Now let $F=F_0+c$ ($c=$ constant) be an arbitrary antiderivative of", + "$f$ on $[a,b]$. Then", + "\\vskip-2pt", + "$$", + "F(b)-F(a)=\\int_a^b f(x)\\,dx+c-\\int_a^a f(x)\\,dx-c=\\int_a^b f(x)\\,dx.", + "$$", + "\\vskip-2.5em" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.3.10", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.11" + ], + "ref_ids": [ + 62, + 63 + ] + } + ], + "ref_ids": [] + }, + { + "id": 66, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.15", + "categories": [], + "title": "Integration by Parts", + "contents": [ + "If $u'$ and $v'$ are integrable on $[a,b],$ then", + "\\begin{equation}\\label{eq:3.3.16}", + "\\int_a^b u(x)v'(x)\\,dx=u(x)v(x)\\bigg|^b_a-\\int_a^b v(x)u'(x)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $u$ and $v$ are continuous", + "on", + "$[a,b]$ (Theorem~\\ref{thmtype:2.3.3}), they", + "are integrable on $[a,b]$. Therefore, Theorems~\\ref{thmtype:3.3.1} and", + "\\ref{thmtype:3.3.6} imply that the function", + "$$", + "(uv)'=u'v+uv'", + "$$", + "is integrable on $[a,b]$, and Theorem~\\ref{thmtype:3.3.12} implies that", + "$$", + "\\int_a^b[u(x)v'(x)+u'(x)v(x)]\\,dx=u(x)v(x)\\bigg|^b_a,", + "$$", + "which implies \\eqref{eq:3.3.16}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.3.3", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.6", + "TRENCH_REAL_ANALYSIS-thmtype:3.3.12" + ], + "ref_ids": [ + 28, + 53, + 58, + 64 + ] + } + ], + "ref_ids": [] + }, + { + "id": 67, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.3.16", + "categories": [], + "title": "Second Mean Value Theorem for Integrals", + "contents": [ + "Suppose that $f'$ is nonnegative and integrable and $g$ is", + "continuous on $[a,b].$ Then", + "\\begin{equation}\\label{eq:3.3.17}", + "\\int_a^b f(x)g(x)\\,dx=f(a)\\int_a^c g(x)\\,dx+f(b)\\int_c^b g(x)\\,dx", + "\\end{equation}", + "for some $c$ in $[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $f$ is differentiable on $[a,b]$, it is continuous on $[a,b]$", + "(Theorem~\\ref{thmtype:2.3.3}).", + "Since $g$ is continuous on $[a,b]$, so is $fg$", + "(Theorem~\\ref{thmtype:2.2.5}). Therefore,", + "Theorem~\\ref{thmtype:3.2.8} implies", + "that the integrals in \\eqref{eq:3.3.17} exist. If", + "\\begin{equation}\\label{eq:3.3.18}", + "G(x)=\\int_a^x g(t)\\,dt,", + "\\end{equation}", + "then $G'(x)=g(x),\\ a0$ and $f(x)\\ge0$ on some subinterval", + "$[a_1,b)$ of $[a,b),$ and", + "\\begin{equation}\\label{eq:3.4.3}", + "\\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=M.", + "\\end{equation}", + "\\begin{alist}", + "\\item % (a)", + "If $00$. Then", + "$$", + "W_f[x_0-h,x_0+h]<\\epsilon", + "$$", + "for some $h>0$, so", + "$$", + "|f(x)-f(x')|<\\epsilon\\mbox{\\quad if\\quad} x_0-h\\le x,x'\\le x_0+h.", + "$$", + " Letting $x'=x_0$, we conclude that", + "$$", + "|f(x)-f(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|0$, there is a", + "$\\delta>0$ such that", + "$$", + "|f(x)-f(x_0)|<\\frac{\\epsilon}{2}\\mbox{\\quad and\\quad} |f(x')-f(x_0)|<", + "\\frac{\\epsilon}{2}", + "$$", + "if $x_0-\\delta\\le x$, $x'\\le x_0+\\delta$. From the triangle", + "inequality,", + "$$", + "|f(x)-f(x')|\\le|f(x)-f(x_0)|+|f(x')-f(x_0)|<\\epsilon,", + "$$", + "so", + "$$", + "W_f[x_0-h,x_0+h]\\le\\epsilon\\mbox{\\quad if\\quad} h<\\delta;", + "$$", + " therefore, $w_f(x_0)=0$.", + "Similar arguments apply if", + "$x_0=a$ or $x_0=b$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 80, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.5.6", + "categories": [], + "title": "", + "contents": [ + "A bounded function $f$ is integrable on a finite interval $[a,b]$ if", + "and only if the set $S$ of discontinuities of $f$ in $[a,b]$ is of", + "Lebesgue measure zero$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Theorem~\\ref{thmtype:3.5.2},", + "$$", + "S=\\set{x\\in [a,b]}{w_f(x)>0}\\negthickspace.", + "$$", + "Since $w_f(x)>0$ if and only if $w_f(x)\\ge1/i$ for some positive", + "integer $i$, we can write", + "\\begin{equation} \\label{eq:3.5.12}", + "S=\\bigcup^\\infty_{i=1} S_i,", + "\\end{equation}", + "where", + "$$", + "S_i=\\set{x\\in [a,b]}{w_f(x)\\ge1/i}.", + "$$", + "Now suppose that $f$ is integrable on $[a,b]$ and $\\epsilon>0$.", + "From Lemma~\\ref{thmtype:3.5.4},", + " each $S_i$ can be covered by a finite number of", + "open intervals $I_{i1}$, $I_{i2}$, \\dots, $I_{in}$ of total length", + "less than", + "$\\epsilon/2^i$. We simply renumber these intervals consecutively;", + "thus,", + "$$", + "I_1,I_2, \\dots=", + "I_{11}, \\dots,I_{1n_1},I_{21}, \\dots,I_{2n_2}, \\dots,", + "I_{i1}, \\dots,I_{in_i}, \\dots.", + "$$", + "Now \\eqref{eq:3.5.8} and \\eqref{eq:3.5.9} hold because of \\eqref{eq:3.5.11} and", + "\\eqref{eq:3.5.12}, and we have shown that the stated condition is", + "necessary for integrability.", + "For sufficiency, suppose that the stated condition holds and", + "$\\epsilon>0$. Then $S$ can be covered by open intervals", + "$I_1,I_2, \\dots$ that satisfy \\eqref{eq:3.5.9}. If $\\rho>0$, then the", + "set", + "$$", + "E_\\rho=\\set{x\\in [a,b]}{w_f(x)\\ge\\rho}", + "$$", + "of Lemma~\\ref{thmtype:3.5.4} is contained in $S$", + "(Theorem~\\ref{thmtype:3.5.2}), and therefore $E_\\rho$ is covered by", + "$I_1,I_2, \\dots$. Since $E_\\rho$ is closed (Lemma~\\ref{thmtype:3.5.4})", + "and bounded, the Heine--Borel theorem implies that $E_\\rho$ is covered", + "by a finite number of intervals from $I_1,I_2, \\dots$. The sum of", + "the lengths of the latter is less than $\\epsilon$, so", + "Lemma~\\ref{thmtype:3.5.4} implies that $f$ is integrable on $[a,b]$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.5.2", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.2", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.4", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.4" + ], + "ref_ids": [ + 79, + 249, + 249, + 79, + 249, + 249 + ] + } + ], + "ref_ids": [] + }, + { + "id": 81, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.2", + "categories": [], + "title": "", + "contents": [ + "The limit of a convergent sequence is unique$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that", + "$$", + "\\lim_{n\\to\\infty}s_n=s\\mbox{\\quad and \\quad}", + "\\lim_{n\\to\\infty}s_n=s'.", + "$$", + "\\vskip5pt", + "\\noindent We must show that $s=s'$.", + "Let $\\epsilon>0$. From Definition~\\ref{thmtype:4.1.1}, there are", + "integers $N_1$ and $N_2$ such that", + "$$", + "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1", + "$$", + "\\vskip5pt", + "\\noindent(because $\\lim_{n\\to\\infty} s_n=s$), and", + "$$", + "|s_n-s'|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_2", + "$$", + "\\newpage", + "\\noindent", + "(because $\\lim_{n\\to\\infty}s_n=s'$). These inequalities both hold if", + "$n\\ge N=\\max (N_1,N_2)$, which implies that", + "\\begin{eqnarray*}", + "|s-s'|\\ar=|(s-s_N)+(s_N-s')|\\\\", + "\\ar\\le |s-s_N|+|s_N-s'|<\\epsilon+\\epsilon=2\\epsilon.", + "\\end{eqnarray*}", + "Since this inequality holds for every $\\epsilon>0$ and $|s-s'|$", + "is independent of $\\epsilon$, we conclude that $|s-s'|=0$; that is,", + "$s=s'$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" + ], + "ref_ids": [ + 324 + ] + } + ], + "ref_ids": [] + }, + { + "id": 82, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.4", + "categories": [], + "title": "", + "contents": [ + "A convergent sequence is bounded$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "By taking $\\epsilon=1$ in \\eqref{eq:4.1.2}, we see that if", + " $\\lim_{n\\to\\infty} s_n=s$, then there is an integer $N$", + "such that", + "$$", + "|s_n-s|<1\\mbox{\\quad if\\quad} n\\ge N.", + "$$", + "Therefore,", + "$$", + "|s_n|=|(s_n-s)+s|\\le|s_n-s|+|s|<1+|s|\\mbox{\\quad if\\quad} n\\ge N,", + "$$", + "and", + "$$", + "|s_n|\\le\\max\\{|s_0|,|s_1|, \\dots,|s_{N-1}|, 1+|s|\\}", + "$$", + "for all $n$, so $\\{s_n\\}$ is bounded." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 83, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + " If $\\{s_n\\}$ is nondecreasing$,$", + "then $\\lim_{n\\to\\infty}s_n=\\sup\\{s_n\\}.$", + "\\item % (b", + "If $\\{s_n\\}$ is nonincreasing$,$ then $\\lim_{n\\to\\infty}s_n=", + "\\inf\\{s_n\\}.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a}. Let $\\beta=\\sup\\{s_n\\}$.", + "If $\\beta<\\infty$, Theorem~\\ref{thmtype:1.1.3}", + "implies that if $\\epsilon>0$ then", + "$$", + "\\beta-\\epsilonb$", + "for some integer $N$. Then $s_n>b$ for $n\\ge N$, so", + "$\\lim_{n\\to\\infty}s_n=\\infty$.", + "We leave the proof of \\part{b}", + "to you (Exercise~\\ref{exer:4.1.8})" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" + ], + "ref_ids": [ + 1, + 324 + ] + } + ], + "ref_ids": [] + }, + { + "id": 84, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.7", + "categories": [], + "title": "", + "contents": [ + " Let $\\lim_{x\\to\\infty} f(x)=L,$", + "where $L$ is in the extended reals$,$ and suppose that", + "$s_n=f(n)$ for large $n.$ Then", + "$$", + "\\lim_{n\\to\\infty}s_n=L.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 85, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.8", + "categories": [], + "title": "", + "contents": [ + " Let", + "\\begin{equation}\\label{eq:4.1.4}", + "\\lim_{n\\to\\infty} s_n=s\\mbox{\\quad and\\quad}\\lim_{n\\to\\infty} t_n=t,", + "\\end{equation}", + "where $s$ and $t$ are finite$.$ Then", + "\\begin{equation}\\label{eq:4.1.5}", + "\\lim_{n\\to\\infty} (cs_n)=cs", + "\\end{equation}", + "if $c$ is a constant$;$", + "\\begin{eqnarray}", + "\\lim_{n\\to\\infty}(s_n+t_n)\\ar=s+t,\\label{eq:4.1.6}\\\\", + "\\lim_{n\\to\\infty}(s_n-t_n)\\ar=s-t, \\label{eq:4.1.7}\\\\", + "\\lim_{n\\to\\infty}(s_nt_n)\\ar=st,\\label{eq:4.1.8}\\\\", + "\\arraytext{and}\\nonumber\\\\", + "\\lim_{n\\to\\infty}\\frac{s_n}{ t_n}\\ar=\\frac{s}{ t}\\label{eq:4.1.9}", + "\\end{eqnarray}", + "if $t_n$ is nonzero for all $n$ and $t\\ne0$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We prove \\eqref{eq:4.1.8} and \\eqref{eq:4.1.9}", + "and leave the rest to you", + "(Exercises~\\ref{exer:4.1.15} and \\ref{exer:4.1.17}). For", + "\\eqref{eq:4.1.8}, we write", + "$$", + "s_nt_n-st=s_nt_n-st_n+st_n-st", + "=(s_n-s)t_n+s(t_n-t);", + "$$", + "\\newpage", + "\\noindent", + "hence,", + "\\begin{equation}\\label{eq:4.1.10}", + "|s_nt_n-st|\\le |s_n-s|\\,|t_n|+|s|\\,|t_n-t|.", + "\\end{equation}", + "Since $\\{t_n\\}$ converges, it is bounded (Theorem~\\ref{thmtype:4.1.4}).", + "Therefore, there is a number $R$ such that $|t_n|\\le R$ for all $n$,", + "and", + "\\eqref{eq:4.1.10} implies that", + "\\begin{equation}\\label{eq:4.1.11}", + "|s_nt_n-st|\\le R|s_n-s|+|s|\\,|t_n-t|.", + "\\end{equation}", + "From \\eqref{eq:4.1.4}, if $\\epsilon>0$ there are integers", + "$N_1$ and $N_2$ such that", + "\\begin{eqnarray}", + "|s_n-s|\\ar<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1 \\label{eq:4.1.12}\\\\", + "\\arraytext{and}\\nonumber\\\\", + "|t_n-t|\\ar<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_2.\\label{eq:4.1.13}", + "\\end{eqnarray}", + "If $N=\\max (N_1,N_2)$, then \\eqref{eq:4.1.12} and \\eqref{eq:4.1.13} both hold", + "when $n\\ge N$, and \\eqref{eq:4.1.11} implies that", + "$$", + "|s_nt_n-st|\\le (R+|s|)\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", + "$$", + "This proves \\eqref{eq:4.1.8}.", + "Now consider \\eqref{eq:4.1.9} in the special case where $s_n=1$ for all", + "$n$ and $t\\ne 0$; thus, we want to show that", + "$$", + "\\lim_{n\\to\\infty}\\frac{1}{ t_n}=\\frac{1}{ t}.", + "$$", + "First, observe that since $\\lim_{n\\to\\infty} t_n=t\\ne0$, there is an", + "integer $M$ such that $|t_n|\\ge |t|/2$ if $n\\ge M$. To see this,", + "we apply Definition~\\ref{thmtype:4.1.1} with $\\epsilon=|t|/2$; thus,", + "there is an integer $M$ such that $|t_n-t|<|t/2|$ if $n\\ge M$.", + "Therefore,", + "$$", + "|t_n|=|t+(t_n-t)|\\ge ||t|-|t_n-t||\\ge\\frac{|t|}{2}\\mbox{\\quad if", + "\\quad} n\\ge M.", + "$$", + " If $\\epsilon>0$, choose $N_0$ so that $|t_n-t|<\\epsilon$", + "if $n\\ge N_0$,", + " and let $N=\\max (N_0,M)$. Then", + "$$", + "\\left|\\frac{1}{ t_n}-\\frac{1}{ t}\\right|=\\frac{|t-t_n|}{", + "|t_n|\\,|t|}\\le\\frac {2", + "\\epsilon}{ |t|^2}\\mbox{\\quad if\\quad} n\\ge N;", + "$$", + "hence, $\\lim_{n\\to\\infty} 1/t_n=1/t$.", + "Now we obtain \\eqref{eq:4.1.9} in the general case from \\eqref{eq:4.1.8}", + "with $\\{t_n\\}$ replaced by $\\{1/t_n\\}$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.4", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.1" + ], + "ref_ids": [ + 82, + 324 + ] + } + ], + "ref_ids": [] + }, + { + "id": 86, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + "If $\\{s_n\\}$ is bounded above and does not diverge to $-\\infty,$ then", + "there is a unique real number $\\overline{s}$ such that$,$ if", + "$\\epsilon>0,$", + "\\begin{equation}\\label{eq:4.1.16}", + "s_n<\\overline{s}+\\epsilon\\mbox{\\quad for large $n$}", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:4.1.17}", + "s_n>\\overline{s}-\\epsilon\\mbox{\\quad for infinitely many", + " $n$}.", + "\\end{equation}", + "\\item % (b)", + "If $\\{s_n\\}$ is bounded below and does not diverge to $\\infty,$ then", + "there is a unique real number $\\underline{s}$ such that$,$ if", + "$\\epsilon", + ">0,$", + "\\begin{equation}\\label{eq:4.1.18}", + "s_n>\\underline{s}-\\epsilon\\mbox{\\quad for large $n$}", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:4.1.19}", + "s_n<\\underline{s}+\\epsilon\\mbox{\\quad for infinitely many", + "$n$}.", + "\\end{equation}", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will prove \\part{a} and leave the proof of \\part{b} to you", + "(Exercise~\\ref{exer:4.1.23}). Since $\\{s_n\\}$ is bounded above,", + "there is a number $\\beta$ such that $s_n<\\beta$ for all", + "$n$. Since $\\{s_n\\}$ does not diverge to $-\\infty$, there is", + "a number $\\alpha$ such that", + "$s_n> \\alpha$ for infinitely many $n$. If we define", + "$$", + "M_k=\\sup\\{s_k,s_{k+1}, \\dots,s_{k+r}, \\dots\\},", + "$$", + "\\newpage", + "\\noindent", + "then $\\alpha\\le M_k\\le\\beta$, so $\\{M_k\\}$ is bounded. Since", + "$\\{M_k\\}$ is nonincreasing (why?), it converges, by", + "Theorem~\\ref{thmtype:4.1.6}. Let", + "\\begin{equation} \\label{eq:4.1.20}", + "\\overline{s}=\\lim_{k\\to\\infty} M_k.", + "\\end{equation}", + "If $\\epsilon>0$, then $M_k<\\overline{s}+\\epsilon$ for large $k$, and", + "since $s_n\\le M_k$ for $n\\ge k$, $\\overline{s}$ satisfies", + "\\eqref{eq:4.1.16}.", + "If \\eqref{eq:4.1.17} were false for some positive", + "$\\epsilon$, there would be an integer $K$ such that", + "$$", + "s_n\\le\\overline{s}-\\epsilon\\mbox{\\quad if\\quad} n\\ge K.", + "$$", + "However, this implies that", + "$$", + "M_k\\le\\overline{s}-\\epsilon\\mbox{\\quad if\\quad} k\\ge K,", + "$$", + "which contradicts \\eqref{eq:4.1.20}. Therefore, $\\overline{s}$", + " has the stated properties.", + "Now we must show that", + "$\\overline{s}$ is the only real number with the stated properties.", + "If $t<\\overline{s}$, the inequality", + "$$", + "s_n t-\\frac{t-\\overline{s}}{2}=\\overline{s}+\\frac{t-\\overline{s}}{", + "2}", + "$$", + "cannot hold for infinitely many $n$, because this would contradict", + "\\eqref{eq:4.1.16} with $\\epsilon=(t-\\overline{s})/2$. Therefore,", + "$\\overline{s}$ is the only real number with the stated properties." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.6" + ], + "ref_ids": [ + 83 + ] + } + ], + "ref_ids": [] + }, + { + "id": 87, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.11", + "categories": [], + "title": "", + "contents": [ + "Every sequence $\\{s_n\\}$ of real numbers has a unique limit", + "superior$,$", + "$\\overline{s},$ and a unique limit inferior$,$ $\\underline{s}$, in the", + "extended reals$,$ and", + "\\begin{equation}\\label{eq:4.1.21}", + "\\underline{s}\\le \\overline{s}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The existence and uniqueness of $\\overline{s}$ and", + "$\\underline{s}$ follow from Theorem~\\ref{thmtype:4.1.9} and", + "Definition~\\ref{thmtype:4.1.10}. If $\\overline{s}$ and $\\underline{s}$ are", + "both finite, then \\eqref{eq:4.1.16} and \\eqref{eq:4.1.18} imply that", + "$$", + "\\underline{s}-\\epsilon<\\overline{s}+\\epsilon", + "$$", + "for every $\\epsilon>0$, which implies \\eqref{eq:4.1.21}. If", + "$\\underline{s}=-\\infty$ or $\\overline{s}=\\infty$, then \\eqref{eq:4.1.21}", + "is obvious. If $\\underline{s}=\\infty$ or $\\overline{s}=-\\infty$, then", + "\\eqref{eq:4.1.21} follows immediately from Definition~\\ref{thmtype:4.1.10}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.10", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.10" + ], + "ref_ids": [ + 86, + 327, + 327 + ] + } + ], + "ref_ids": [] + }, + { + "id": 88, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.12", + "categories": [], + "title": "", + "contents": [ + "If $\\{s_n\\}$ is a sequence of real numbers, then", + "\\begin{equation}\\label{eq:4.1.22}", + "\\lim_{n\\to\\infty} s_n=s", + "\\end{equation}", + "if and only if", + "\\begin{equation}\\label{eq:4.1.23}", + "\\limsup_{n\\to\\infty}s_n=\\liminf_{n\\to\\infty} s_n=s.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $s=\\pm\\infty$, the equivalence of \\eqref{eq:4.1.22} and", + "\\eqref{eq:4.1.23} follows immediately from their definitions. If", + "$\\lim_{n\\to\\infty}s_n=s$ (finite), then Definition~\\ref{thmtype:4.1.1}", + "implies that \\eqref{eq:4.1.16}--\\eqref{eq:4.1.19} hold with $\\overline{s}$ and $\\underline{s}$ replaced by", + "$s$. Hence, \\eqref{eq:4.1.23} follows from the uniqueness of", + "$\\overline{s}$ and $\\underline{s}$. For the converse, suppose that", + "$\\overline{s}=\\underline{s}$ and let $s$ denote their common value.", + "Then \\eqref{eq:4.1.16} and \\eqref{eq:4.1.18} imply that", + "$$", + "s-\\epsilon0,$ there is an integer $N$ such that", + "\\begin{equation}\\label{eq:4.1.24}", + "|s_n-s_m|<\\epsilon\\mbox{\\quad if\\quad} m,n\\ge N.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $\\lim_{n\\to\\infty}s_n=s$ and $\\epsilon>0$.", + "By Definition~\\ref{thmtype:4.1.1}, there is an integer $N$ such that", + "$$", + "|s_r-s|<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} r\\ge N.", + "$$", + "Therefore,", + "$$", + "|s_n-s_m|=|(s_n-s)+(s-s_m)|\\le |s_n-s|+|s-s_m|<\\epsilon", + "\\mbox{\\quad if\\quad} n,m\\ge N.", + "$$", + "Therefore, the stated condition is necessary for convergence of", + "$\\{s_n\\}$. To see that it is sufficient, we first observe that it", + "implies that $\\{s_n\\}$ is bounded (Exercise~\\ref{exer:4.1.27}), so", + "$\\overline{s}$ and $\\underline{s}$ are finite", + "(Theorem~\\ref{thmtype:4.1.9}).", + "Now suppose that $\\epsilon>0$ and $N$ satisfies \\eqref{eq:4.1.24}. From", + "\\eqref{eq:4.1.16} and \\eqref{eq:4.1.17},", + "\\begin{equation}\\label{eq:4.1.25}", + "|s_n-\\overline{s}|<\\epsilon,", + "\\end{equation}", + "for some integer $n>N$ and, from \\eqref{eq:4.1.18} and \\eqref{eq:4.1.19},", + "\\begin{equation}\\label{eq:4.1.26}", + "|s_m-\\underline{s}|<\\epsilon", + "\\end{equation}", + "for some integer $m>N$. Since", + "\\begin{eqnarray*}", + "|\\overline{s}-\\underline{s}|\\ar=|(\\overline{s}-s_n)+", + "(s_n-s_m)+(s_m-\\underline{s})|\\\\", + "\\ar\\le |\\overline{s}-s_n|+|s_n-s_m|+|s_m-\\underline{s}|,", + "\\end{eqnarray*}", + "\\eqref{eq:4.1.24}--\\eqref{eq:4.1.26} imply that", + "$$", + "|\\overline{s}-\\underline{s}|<3\\epsilon.", + "$$", + "Since $\\epsilon$ is an arbitrary positive number, this implies that", + "$\\overline{s}=\\underline{s}$, so $\\{s_n\\}$ converges, by", + "Theorem~\\ref{thmtype:4.1.12}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.1", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.9", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.12" + ], + "ref_ids": [ + 324, + 86, + 88 + ] + } + ], + "ref_ids": [] + }, + { + "id": 90, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.2", + "categories": [], + "title": "", + "contents": [ + "If", + "\\begin{equation}\\label{eq:4.2.1}", + "\\lim_{n\\to\\infty}s_n=s\\quad (-\\infty\\le s\\le\\infty),", + "\\end{equation}", + "then", + "\\begin{equation}\\label{eq:4.2.2}", + "\\lim_{k\\to\\infty} s_{n_k}=s", + "\\end{equation}", + "for every subsequence $\\{s_{n_k}\\}$ of $\\{s_n\\}.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We consider the case where $s$ is finite and leave the rest to you", + "(Exercise~\\ref{exer:4.2.4}). If \\eqref{eq:4.2.1} holds and $\\epsilon>0$,", + "there is an integer $N$ such that", + "$$", + "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", + "$$", + "Since $\\{n_k\\}$ is an increasing sequence, there is an integer $K$", + "such that", + "$n_k\\ge N$ if $k\\ge K$. Therefore,", + "$$", + "|s_{n_k}-L|<\\epsilon\\mbox{\\quad if\\quad} k\\ge K,", + "$$", + "which implies \\eqref{eq:4.2.2}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 91, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.3", + "categories": [], + "title": "", + "contents": [ + " If $\\{s_n\\}$ is monotonic and has a", + "subsequence $\\{s_{n_k}\\}$ such that", + "$$", + "\\lim_{k\\to\\infty} s_{n_k}=s\\quad (-\\infty\\le s\\le\\infty),", + "$$", + "then", + "$$", + "\\lim_{n\\to\\infty} s_n=s.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We consider the case where $\\{s_n\\}$ is nondecreasing and leave", + "the rest to you (Exercise~\\ref{exer:4.2.6}). Since $\\{s_{n_k}\\}$ is also", + "nondecreasing in this case, it suffices to show that", + "\\begin{equation}\\label{eq:4.2.3}", + "\\sup\\{s_{n_k}\\}=\\sup\\{s_n\\}", + "\\end{equation}", + "and then apply Theorem~\\ref{thmtype:4.1.6}\\part{a}. Since the", + "set of terms of", + "$\\{s_{n_k}\\}$ is contained in the set of terms of $\\{s_n\\}$,", + "\\begin{equation} \\label{eq:4.2.4}", + "\\sup\\{s_n\\}\\ge\\sup\\{s_{n_k}\\}.", + "\\end{equation}", + "Since $\\{s_n\\}$ is nondecreasing, there is for every $n$ an integer", + "$n_k$ such that $s_n\\le s_{n_k}$. This implies that", + "$$", + "\\sup\\{s_n\\}\\le\\,\\sup\\{s_{n_k}\\}.", + "$$", + "This and \\eqref{eq:4.2.4} imply \\eqref{eq:4.2.3}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.6" + ], + "ref_ids": [ + 83 + ] + } + ], + "ref_ids": [] + }, + { + "id": 92, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.4", + "categories": [], + "title": "", + "contents": [ + "A point $\\overline{x}$ is a limit", + "point of a set $S$ if and only if there is a sequence $\\{x_n\\}$ of points", + "in $S$ such that $x_n\\ne\\overline{x}$ for $n\\ge 1,$ and", + "$$", + "\\lim_{n\\to\\infty}x_n=\\overline{x}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For sufficiency, suppose that the stated condition holds.", + "Then, for each $\\epsilon>0$, there is an integer $N$ such", + "that $0<|x_n-x|<\\epsilon$ if $n\\ge N$. Therefore, every", + "$\\epsilon$-neighborhood of $\\overline{x}$ contains infinitely many", + "points of $S$. This means that $\\overline{x}$ is a limit point of $S$.", + "For necessity, let $\\overline{x}$ be a limit point of $S$. Then,", + "for every integer $n\\ge1$,", + "the interval $(\\overline{x}-1/n,\\overline{x}+1/n)$", + "contains", + "a point $x_n\\ (\\ne\\overline{x})$ in $S$. Since", + "$|x_m-\\overline{x}|\\le1/n$ if $m\\ge n$, $\\lim_{n\\to\\infty}x_n=", + "\\overline{x}$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 93, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.5", + "categories": [], + "title": "", + "contents": [ + "\\vspace*{3pt}", + "\\begin{alist}", + "\\item % (a)", + " If $\\{x_n\\}$ is bounded$,$ then", + "$\\{x_n\\}$ has a convergent subsequence$.$", + "\\vspace*{3pt}", + "\\item % (b)", + " If $\\{x_n\\}$ is unbounded above$,$", + " then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that", + "$$", + "\\lim_{k\\to\\infty} x_{n_k}=\\infty.", + "$$", + "\\vspace*{3pt}", + "\\item % (c)", + " If $\\{x_n\\}$ is unbounded", + "below$,$ then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that", + "$$", + "\\lim_{k\\to\\infty} x_{n_k}=-\\infty.", + "$$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We prove \\part{a} and leave \\part{b} and \\part{c} to you", + "(Exercise~\\ref{exer:4.2.7}). Let", + "$S$ be the set of distinct numbers that occur as terms of $\\{x_n\\}$.", + "(For example, if $\\{x_n\\}=\\{(-1)^n\\}$, $S=\\{1,-1\\}$; if", + "$\\{x_n\\}=\\{1,\\frac{1}{2}, 1, \\frac{1}{3}, \\dots, 1, 1/n, \\dots\\}$,", + "$S=\\{1,\\frac{1}{2}, \\dots, 1/n, \\dots\\}$.) If $S$ contains only finitely", + "many points, then some $\\overline{x}$ in $S$ occurs infinitely often", + "in $\\{x_n\\}$; that is, $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such", + "that $x_{n_k}=\\overline{x}$ for all $k$. Then", + "$\\lim_{k\\to\\infty}", + "x_{n_k}=\\overline{x}$, and we are finished in this case.", + "If $S$ is infinite, then, since $S$ is bounded (by assumption), the", + "Bolzano--Weierstrass theorem (Theorem~\\ref{thmtype:1.3.8})", + "implies that", + "$S$ has a limit point", + "$\\overline{x}$. From Theorem~\\ref{thmtype:4.2.4}, there is a sequence of", + "points $\\{y_j\\}$ in $S$, distinct from $\\overline{x}$, such that", + "\\begin{equation}\\label{eq:4.2.5}", + "\\lim_{j\\to\\infty} y_j=\\overline{x}.", + "\\end{equation}", + "Although each $y_j$ occurs as a term of $\\{x_n\\}$, $\\{y_j\\}$ is", + "not necessarily a subsequence of $\\{x_n\\}$, because if we write", + "$$", + "y_j=x_{n_j},", + "$$", + "there is no reason to expect that $\\{n_j\\}$ is an increasing sequence", + "as required in Definition~\\ref{thmtype:4.2.1}. However, it is always", + "possible to pick a subsequence $\\{n_{j_k}\\}$ of $\\{n_j\\}$ that is", + "increasing, and then the sequence $\\{y_{j_k}\\}=\\{s_{n_{j_k}}\\}$ is a", + "subsequence of both $\\{y_j\\}$ and $\\{x_n\\}$. Because of \\eqref{eq:4.2.5}", + "and Theorem~\\ref{thmtype:4.2.2} this subsequence converges", + "to~$\\overline{x}$.", + "\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.3.8", + "TRENCH_REAL_ANALYSIS-thmtype:4.2.4", + "TRENCH_REAL_ANALYSIS-thmtype:4.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:4.2.2" + ], + "ref_ids": [ + 12, + 92, + 328, + 90 + ] + } + ], + "ref_ids": [] + }, + { + "id": 94, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.6", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be defined on a closed interval $[a,b]$ containing", + "$\\overline{x}.$ Then $f$ is continuous at $\\overline{x}$", + "$($from the right if $\\overline{x}=a,$ from the left if", + "$\\overline{x}=b$$)$ if and only if", + "\\begin{equation}\\label{eq:4.2.6}", + "\\lim_{n\\to\\infty} f(x_n)=f(\\overline{x})", + "\\end{equation}", + "whenever $\\{x_n\\}$ is a sequence of points in $[a,b]$ such that", + "\\begin{equation}\\label{eq:4.2.7}", + "\\lim_{n\\to\\infty} x_n=\\overline{x}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Assume that $a<\\overline{x}0$, there is a", + "$\\delta> 0$ such that", + "\\begin{equation} \\label{eq:4.2.8}", + "|f(x)-f(\\overline{x})|<\\epsilon\\mbox{\\quad if\\quad} |x-\\overline{x}|", + "<\\delta.", + "\\end{equation}", + "From \\eqref{eq:4.2.7}, there is an integer $N$ such that", + "$|x_n-\\overline{x}|<\\delta$", + " if $n\\ge N$. This and \\eqref{eq:4.2.8} imply that", + "$|f(x_n)-f(\\overline{x})|<\\epsilon$ if $n\\ge N$. This implies", + "\\eqref{eq:4.2.6}, which shows that the stated condition is necessary.", + "For sufficiency, suppose that $f$ is discontinuous at $\\overline{x}$.", + "Then there is an $\\epsilon_0>0$ such that, for each positive integer", + "$n$, there is a point $x_n$ that satisfies the inequality", + "$$", + "|x_n-\\overline{x}|<\\frac{1}{ n}", + "$$", + "\\newpage", + "\\noindent", + "while", + "$$", + "|f(x_n)-f(\\overline{x})|\\ge\\epsilon_0.", + "$$", + "The sequence $\\{x_n\\}$ therefore satisfies \\eqref{eq:4.2.7}, but not", + "\\eqref{eq:4.2.6}. Hence, the stated condition cannot hold if $f$ is", + "discontinuous at $\\overline{x}$. This proves sufficiency." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 95, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.2.7", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a closed", + "interval $[a,b],$ then $f$ is bounded on $[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The proof is by contradiction.", + "If $f$ is not bounded on $[a,b]$, there is for each positive", + "integer $n$ a point $x_n$ in $[a,b]$ such that", + "$|f(x_n)|>n$. This implies that", + "\\begin{equation}\\label{eq:4.2.9}", + "\\lim_{n\\to\\infty}|f(x_n)|=\\infty.", + "\\end{equation}", + "Since $\\{x_n\\}$ is bounded, $\\{x_n\\}$ has a convergent subsequence", + "$\\{x_{n_k}\\}$ (Theorem~\\ref{thmtype:4.2.5}\\part{a}). If", + "$$", + "\\overline{x}=\\lim_{k\\to\\infty} x_{n_k},", + "$$", + "then $\\overline{x}$ is a limit point of $[a,b]$, so", + "$\\overline{x}\\in [a,b]$. If $f$ is continuous on $[a,b]$, then", + "$$", + "\\lim_{k\\to\\infty} f(x_{n_k})=f(\\overline{x})", + "$$", + "by Theorem~\\ref{thmtype:4.2.6}, so", + "$$", + "\\lim_{k\\to\\infty} |f(x_{n_k})|=|f(\\overline{x})|", + "$$", + "(Exercise~\\ref{exer:4.1.6}), which contradicts", + "\\eqref{eq:4.2.9}.", + "Therefore, $f$ cannot be both continuous and unbounded on $[a,b]$" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.2.5", + "TRENCH_REAL_ANALYSIS-thmtype:4.2.6" + ], + "ref_ids": [ + 93, + 94 + ] + } + ], + "ref_ids": [] + }, + { + "id": 96, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.2", + "categories": [], + "title": "", + "contents": [ + "The sum of a convergent series is unique$.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 97, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.3", + "categories": [], + "title": "", + "contents": [ + "Let", + "$$", + "\\sum_{n=k}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=k}^\\infty b_n=B,", + "$$", + "where $A$ and $B$ are finite$.$ Then", + "$$", + "\\sum_{n=k}^\\infty (ca_n)=cA", + "$$", + "if $c$ is a constant$,$", + "$$", + "\\sum_{n=k}^\\infty (a_n+b_n)=A+B,", + "$$", + "and", + "$$", + "\\sum_{n=k}^\\infty (a_n-b_n)=A-B.", + "$$", + "These relations also hold if one or both of $A$ and $B$ is infinite,", + "provided that the right sides are not indeterminate$.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 98, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.5", + "categories": [], + "title": "Cauchy's Convergence Criterion for Series", + "contents": [ + "A series $\\sum a_n$ converges if and only if for every", + "$\\epsilon>0$", + "there is an integer $N$ such that", + "\\begin{equation}\\label{eq:4.3.3}", + "|a_n+a_{n+1}+\\cdots+a_m|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "In terms of the partial sums $\\{A_n\\}$ of $\\sum a_n$,", + "$$", + "a_n+a_{n+1}+\\cdots+a_m=A_m-A_{n-1}.", + "$$", + "Therefore, \\eqref{eq:4.3.3} can be written as", + "$$", + "|A_m-A_{n-1}|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", + "$$", + "Since $\\sum a_n$ converges if and only if $\\{A_n\\}$ converges,", + "Theorem~\\ref{thmtype:4.1.13} implies the conclusion." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" + ], + "ref_ids": [ + 89 + ] + } + ], + "ref_ids": [] + }, + { + "id": 99, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.8", + "categories": [], + "title": "", + "contents": [ + "If $a_n\\ge0$ for $n\\ge k,$ then $\\sum a_n$ converges if its partial", + "sums are bounded$,$ or diverges to $\\infty$ if they are not$.$ These", + "are the only possibilities and$,$ in either case$,$", + "$$", + "\\sum_{n=k}^\\infty a_n =\\,\\sup\\set{A_n}{n\\ge k}\\negthickspace,", + "$$", + "where", + "$$", + "A_n=a_k+a_{k+1}+\\cdots+a_n,\\quad n\\ge k.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $A_n=A_{n-1}+a_n$ and $a_n\\ge0$ $(n\\ge k)$, the sequence", + "$\\{A_n\\}$ is nondecreasing, so the conclusion follows from", + "Theorem~\\ref{thmtype:4.1.6}\\part{a} and", + "Definition~\\ref{thmtype:4.3.1}.", + "\\newline\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.1" + ], + "ref_ids": [ + 83, + 329 + ] + } + ], + "ref_ids": [] + }, + { + "id": 100, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.9", + "categories": [], + "title": "The Comparison Test", + "contents": [ + "Suppose that", + "\\begin{equation}\\label{eq:4.3.5}", + "0\\le a_n\\le b_n,\\quad n\\ge k.", + "\\end{equation}", + "Then", + "\\begin{alist}", + "\\item % (a)", + " $\\sum a_n<\\infty$ if $\\sum b_n<\\infty$$.$", + "\\item % (b)", + " $\\sum b_n=\\infty$ if $\\sum a_n=\\infty.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} If", + "$$", + "A_n=a_k+a_{k+1}+\\cdots+a_n\\mbox{\\quad and\\quad} B_n=b_k+", + "b_{k+1}+\\cdots+b_n,\\quad n\\ge k,", + "$$", + "then, from \\eqref{eq:4.3.5},", + "\\begin{equation}\\label{eq:4.3.6}", + "A_n\\le B_n.", + "\\end{equation}", + "Now we use Theorem~\\ref{thmtype:4.3.8}.", + "If $\\sum b_n<\\infty$, then $\\{B_n\\}$ is bounded above", + " and \\eqref{eq:4.3.6} implies that $\\{A_n\\}$ is", + "also; therefore, $\\sum a_n<\\infty$.", + "On the other hand, if", + " $\\sum a_n=\\infty$, then $\\{A_n\\}$ is unbounded above", + " and \\eqref{eq:4.3.6} implies that $\\{B_n\\}$ is", + "also; therefore, $\\sum b_n~=~\\infty$.", + "\\vspace*{4pt}", + "We leave it to you to show that \\part{a} implies \\part{b}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.8" + ], + "ref_ids": [ + 99 + ] + } + ], + "ref_ids": [] + }, + { + "id": 101, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.10", + "categories": [], + "title": "The Integral Test", + "contents": [ + "Let", + "\\begin{equation}\\label{eq:4.3.7}", + "c_n=f(n),\\quad n\\ge k,", + "\\end{equation}", + "where $f$ is positive$,$ nonincreasing$,$ and locally integrable on", + "$[k,\\infty).$", + "Then", + "\\begin{equation}\\label{eq:4.3.8}", + "\\sum c_n<\\infty", + "\\end{equation}", + "if and only if", + "\\begin{equation}\\label{eq:4.3.9}", + "\\int^\\infty_k f(x)\\,dx<\\infty.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We first observe that \\eqref{eq:4.3.9} holds if and only if", + "\\begin{equation}\\label{eq:4.3.10}", + "\\sum_{n=k}^\\infty \\int^{n+1}_n f(x)\\,dx<\\infty", + "\\end{equation}", + "(Exercise~\\ref{exer:4.3.9}), so it is enough to show that \\eqref{eq:4.3.8}", + "holds if and only if \\eqref{eq:4.3.10} does. From \\eqref{eq:4.3.7} and the", + "assumption that $f$ is nonincreasing,", + "$$", + "c_{n+1}=f(n+1)\\le f(x)\\le f(n)=c_n,\\quad n\\le x\\le n+1,\\quad n\\ge k.", + "$$", + "Therefore,", + "$$", + "c_{n+1}=\\int^{n+1}_n c_{n+1}\\,dx\\le\\int^{n+1}_n f(x)\\,dx\\le", + "\\int^{n+1}_n c_n\\,dx=c_n,\\quad n\\ge k", + "$$", + "(Theorem~\\ref{thmtype:3.3.4}). From the first inequality and", + "Theorem~\\ref{thmtype:4.3.9}\\part{a} with $a_n=c_{n+1}$ and", + "$b_n=\\int^{n+1}_n", + "f(x)\\,dx$, \\eqref{eq:4.3.10} implies that $\\sum c_{n+1}<\\infty$, which is", + "equivalent to \\eqref{eq:4.3.8}. From the second inequality and", + "Theorem~\\ref{thmtype:4.3.9}\\part{a} with $a_n=\\int^{n+1}_n f(x)\\,dx$ and", + "$b_n=c_n$, \\eqref{eq:4.3.8} implies \\eqref{eq:4.3.10}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.3.4", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.9", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.9" + ], + "ref_ids": [ + 56, + 100, + 100 + ] + } + ], + "ref_ids": [] + }, + { + "id": 102, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.11", + "categories": [], + "title": "", + "contents": [ + "Suppose that $a_n\\ge0$ and $b_n>0$ for $n\\ge k.$ Then", + "\\begin{alist}", + "\\item % (a)", + "$\\dst{\\sum a_n<\\infty\\mbox{\\quad if\\quad}\\sum b_n<", + "\\infty\\mbox{\\quad and\\quad}\\limsup_{n\\to\\infty} a_n/b_n<\\infty}.$", + "\\item % (b)", + " $\\dst{\\sum a_n=\\infty\\mbox{\\quad if\\quad}\\sum b_n=", + "\\infty\\mbox{\\quad and\\quad}\\liminf_{n\\to\\infty} a_n/b_n>0}.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} If", + "$\\limsup_{n\\to\\infty} a_n/b_n<\\infty$, then $\\{a_n/b_n\\}$ is", + "bounded, so there is a constant $M$ and an integer $k$ such that", + "$$", + "a_n\\le Mb_n,\\quad n\\ge k.", + "$$", + "Since $\\sum b_n<\\infty$, Theorem~\\ref{thmtype:4.3.3} implies that $\\sum", + "(Mb_n)< \\infty$. Now", + "$\\sum a_n<\\infty$, by the comparison test.", + "\\part{b}", + "If", + "$\\liminf_{n\\to\\infty} a_n/b_n>0$,", + " there is a constant $m$ and an integer $k$ such that", + "$$", + "a_n\\ge mb_n,\\quad n\\ge k.", + "$$", + "Since $\\sum b_n=\\infty$, Theorem~\\ref{thmtype:4.3.3} implies that $\\sum", + "(mb_n)= \\infty$. Now", + "$\\sum a_n=\\infty$, by the comparison test." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.3", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.3" + ], + "ref_ids": [ + 97, + 97 + ] + } + ], + "ref_ids": [] + }, + { + "id": 103, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", + "categories": [], + "title": "", + "contents": [ + "Suppose that $a_n>0,$ $b_n>0,$ and", + "\\begin{equation}\\label{eq:4.3.12}", + "\\frac{a_{n+1}}{ a_n}\\le \\frac{b_{n+1}}{ b_n}.", + "\\end{equation}", + "Then", + "\\begin{alist}", + "\\item % (a)", + " $\\sum a_n<\\infty$ if $\\sum b_n<\\infty.$", + "\\item % (b)", + " $\\sum b_n=\\infty$ if $\\sum a_n=\\infty.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Rewriting \\eqref{eq:4.3.12} as", + "$$", + "\\frac{a_{n+1}}{ b_{n+1}}\\le \\frac{a_n}{ b_n},", + "$$", + "we see that $\\{a_n/b_n\\}$ is nonincreasing. Therefore,", + "$\\limsup_{n \\to\\infty} a_n/b_n<\\infty$, and", + "Theorem~\\ref{thmtype:4.3.11}\\part{a} implies \\part{a}.", + "To prove", + "\\part{b}, suppose that $\\sum a_n=\\infty$. Since $\\{a_n/b_n\\}$", + "is nonincreasing,", + " there is a number $\\rho$", + "such that $b_n\\ge \\rho a_n$ for large $n$. Since $\\sum (\\rho", + "a_n)=\\infty$ if $\\sum a_n=\\infty$, Theorem~\\ref{thmtype:4.3.9}\\part{b}", + "(with $a_n$ replaced by $\\rho a_n$)", + "implies that $\\sum b_n=\\infty$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.11", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.9" + ], + "ref_ids": [ + 102, + 100 + ] + } + ], + "ref_ids": [] + }, + { + "id": 104, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.14", + "categories": [], + "title": "The Ratio Test", + "contents": [ + "Suppose that $a_n>0$ for $n\\ge k.$ Then", + "\\vspace*{5pt}", + "\\begin{alist}", + "\\vspace*{5pt}", + "\\item % (a)", + "$\\sum a_n<\\infty$ if\\,", + "$\\limsup_{n\\to\\infty} a_{n+1}/a_n<1.$", + "\\vspace*{5pt}", + "\\item % (b)", + " $\\sum a_n=\\infty$ if\\,", + "$\\liminf_{n\\to\\infty} a_{n+1}/a_n>1.$", + "\\end{alist}", + "\\vspace*{5pt}", + "\\noindent If", + "\\begin{equation}\\label{eq:4.3.13}", + "\\liminf_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}\\le1\\le", + "\\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n},", + "\\end{equation}", + "then the test is inconclusive$;$ that is$,$ $\\sum a_n$ may converge", + "or diverge$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} If", + "$$", + "\\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}<1,", + "$$", + "there is a number $r$ such that $01,", + "$$", + " there is a number $r$ such that $r>1$ and", + "$$", + "\\frac{a_{n+1}}{ a_n}>r", + "$$", + "for $n$ sufficiently large. This can be rewritten as", + "$$", + "\\frac{a_{n+1}}{ a_n}>\\frac{r^{n+1}}{ r^n}.", + "$$", + "Since $\\sum r^n=\\infty$,", + "Theorem~\\ref{thmtype:4.3.13}\\part{b} with $a_n=r^n$ implies that $\\sum", + "b_n=\\infty$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" + ], + "ref_ids": [ + 103, + 103 + ] + } + ], + "ref_ids": [] + }, + { + "id": 105, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.16", + "categories": [], + "title": "", + "contents": [ + "Suppose that $a_n>0$ for large $n.$ Let", + "$$", + "M=\\limsup_{n\\to\\infty} n\\left(\\frac{a_{n+1}}{ a_n}-", + "1\\right)\\mbox{\\quad and\\quad} m=\\liminf_{n\\to\\infty} n", + "\\left(\\frac{a_{n+1}}{ a_n}-1\\right).", + "$$", + "Then", + "\\begin{alist}", + "\\item % (a)", + " $\\sum a_n<\\infty$ if $M<-1.$", + "\\item % (b)", + " $\\sum a_n=\\infty$ if $m>-1.$", + "\\end{alist}", + "The test is inconclusive if $m\\le-1\\le M.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a}", + "We need the inequality", + "\\begin{equation}\\label{eq:4.3.15}", + "\\frac{1}{(1+x)^p}>1-px,\\quad x>0,\\ p>0.", + "\\end{equation}", + "This follows from Taylor's theorem", + "(Theorem~\\ref{thmtype:2.5.4}), which implies that", + "$$", + "\\frac{1}{(1+x)^p}=1-px+\\frac{1}{2}\\frac{p(p+1)}{(1+c)^{p+2}}x^2,", + "$$", + "where $00$,", + "this implies \\eqref{eq:4.3.15}.", + "Now suppose that $M<-p<-1$. Then there is an integer $k$ such that", + "$$", + "n\\left(\\frac{a_{n+1}}{ a_n}-1\\right)<-p,\\quad n\\ge k,", + "$$", + "so", + "$$", + "\\frac{a_{n+1}}{ a_n}<1-\\frac{p}{ n},\\quad n\\ge k.", + "$$", + "Hence,", + "$$", + "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(1+1/n)^p},\\quad n\\ge k,", + "$$", + "as can be seen by letting $x=1/n$ in \\eqref{eq:4.3.15}. From this,", + "$$", + "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(n+1)^p}\\bigg/\\frac{1}{ n^p},\\quad n\\ge k.", + "$$", + " Since $\\sum 1/n^p<\\infty$ if $p>1$,", + " Theorem~\\ref{thmtype:4.3.13}\\part{a} implies that", + " $\\sum a_n<\\infty$.", + "\\part{b} Here we need the inequality", + "\\begin{equation}\\label{eq:4.3.16}", + "(1-x)^q<1-qx,\\quad 0-q,\\quad n\\ge k,", + "$$", + "so", + "$$", + "\\frac{a_{n+1}}{ a_n}\\ge1-\\frac{q}{ n},\\quad n\\ge k.", + "$$", + "If $q\\le0$, then $\\sum a_n=\\infty$, by Corollary~\\ref{thmtype:4.3.6}.", + "Hence, we may assume that $0\\left(1-\\frac{1}{ n}\\right)^q,\\quad n\\ge k,", + "$$", + "\\newpage", + "\\noindent", + "as can be seen by setting $x=1/n$ in \\eqref{eq:4.3.16}. Hence,", + "$$", + "\\frac{a_{n+1}}{ a_n}>\\frac{1}{ n^q}\\bigg/\\frac{1}{(n-1)^q},\\quad n\\ge k.", + "$$", + " Since $\\sum 1/n^q=\\infty$ if $q<1$,", + " Theorem~\\ref{thmtype:4.3.13}\\part{b} implies that", + " $\\sum a_n=\\infty$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.6", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" + ], + "ref_ids": [ + 42, + 103, + 277, + 103 + ] + } + ], + "ref_ids": [] + }, + { + "id": 106, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", + "categories": [], + "title": "Cauchy's Root Test", + "contents": [ + "If $a_n\\ge 0$ for $n\\ge k,$ then", + "\\begin{alist}", + "\\item % (a)", + " $\\sum a_n<\\infty$ if", + "$\\limsup_{n\\to\\infty} a^{1/n}_n<1.$", + "\\item % (b)", + " $\\sum a_n=\\infty$ if", + "$\\limsup_{n\\to\\infty} a^{1/n}_n>1.$", + "\\end{alist}", + "The test is inconclusive if $\\limsup_{n\\to\\infty} a^{1/n}_n=", + "1.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a} If $\\limsup_{n\\to\\infty}a^{1/n}_n<1$, there is an", + " $r$", + "such that $01$, then $a^{1/n}_n>1$", + "for infinitely many values of $n$,", + "so $\\sum a_n=\\infty$, by", + "Corollary~\\ref{thmtype:4.3.6}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" + ], + "ref_ids": [ + 277 + ] + } + ], + "ref_ids": [] + }, + { + "id": 107, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.19", + "categories": [], + "title": "", + "contents": [ + "absolutely$,$ then $\\sum a_n$ converges$.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 108, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.20", + "categories": [], + "title": "Dirichlet's Test for Series", + "contents": [ + "The series $\\sum ^\\infty_{n=k} a_nb_n$ converges if $\\lim_{n\\to\\infty}", + "a_n= 0,$", + "\\begin{equation}\\label{eq:4.3.18}", + "\\sum |a_{n+1}-a_n|<\\infty,", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:4.3.19}", + "|b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k,", + "\\end{equation}", + "for some constant $M.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The proof is similar to the proof of Dirichlet's test for integrals.", + "Define", + "$$", + "B_n=b_k+b_{k+1}+\\cdots+b_n,\\quad n\\ge k", + "$$", + "and consider the partial sums of $\\sum_{n=k}^\\infty a_nb_n$:", + "\\begin{equation}\\label{eq:4.3.20}", + "S_n=a_kb_k+a_{k+1}b_{k+1}+\\cdots+a_nb_n,\\quad n\\ge k.", + "\\end{equation}", + "By substituting", + "$$", + "b_k=B_k\\mbox{\\quad and\\quad} b_n=B_n-B_{n-1},\\quad n\\ge k+1,", + "$$", + "into \\eqref{eq:4.3.20}, we obtain", + "$$", + "S_n=a_kB_k+a_{k+1}(B_{k+1}-B_k)+\\cdots+a_n(B_n-B_{n-1}),", + "$$", + "which we rewrite as", + "\\begin{equation}\\label{eq:4.3.21}", + "\\begin{array}{rcl}", + "S_n\\ar=(a_k-a_{k+1})B_k+(a_{k+1}-a_{k+2})B_{k+1}+\\cdots\\\\", + "\\ar{}+\\,(a_{n-1}-a_n)B_{n-1}+a_nB_n.", + "\\end{array}", + "\\end{equation}", + "\\newpage", + "\\noindent", + "(The procedure that led from \\eqref{eq:4.3.20} to \\eqref{eq:4.3.21} is called", + "{\\it summation by parts\\/}. It is analogous", + "to integration by parts.) Now \\eqref{eq:4.3.21} can be viewed as", + "\\begin{equation}\\label{eq:4.3.22}", + "S_n=T_{n-1}+a_nB_n,", + "\\end{equation}", + "where", + "$$", + "T_{n-1}=(a_k-a_{k+1})B_k+(a_{k+1}-a_{k+2})", + "B_{k+1}+\\cdots+(a_{n-1}-a_n)B_{n-1};", + "$$", + "that is, $\\{T_n\\}$ is the sequence of partial sums of the series", + "\\begin{equation}\\label{eq:4.3.23}", + "\\sum_{j=k}^\\infty (a_j-a_{j+1})B_j.", + "\\end{equation}", + "Since", + "$$", + "|(a_j-a_{j+1})B_j|\\le M|a_j-a_{j+1}|", + "$$", + "from \\eqref{eq:4.3.19}, the comparison test and \\eqref{eq:4.3.18} imply that", + "the series \\eqref{eq:4.3.23} converges absolutely.", + "Theorem~\\ref{thmtype:4.3.19}", + "now implies that $\\{T_n\\}$ converges. Let $T=\\lim_{n\\to\\infty}T_n$.", + "Since $\\{B_n\\}$ is bounded and $\\lim_{n\\to \\infty}a_n=0$, we infer", + "from \\eqref{eq:4.3.22} that", + "$$", + "\\lim_{n\\to\\infty} S_n=\\lim_{n\\to\\infty}T_{n-1}+\\lim_{n\\to", + "\\infty}a_nB_n=T+0=T.", + "$$", + "Therefore, $\\sum a_nb_n$ converges." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.19" + ], + "ref_ids": [ + 107 + ] + } + ], + "ref_ids": [] + }, + { + "id": 109, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.23", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\sum_{n=k}^\\infty a_n=A,$ where $-\\infty \\le A\\le\\infty.$ Let", + "$\\{n_j\\}_1^\\infty$ be an increasing sequence of integers, with $n_1\\ge", + "k$. Define", + "\\begin{eqnarray*}", + "b_1\\ar=a_k+\\cdots+a_{n_1},\\\\", + "b_2\\ar=a_{{n_1}+1}+\\cdots+a_{n_2},\\\\", + "&\\vdots\\\\", + "b_r\\ar=a_{n_{r-1}+1}+\\cdots+a_{n_r}.", + "\\end{eqnarray*}", + "Then", + "$$", + "\\sum_{j=1}^\\infty b_{n_j}=A.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $T_r$ is the $r$th partial sum of $\\sum_{j=1}^\\infty", + "b_{n_j}$ and $\\{A_n\\}$ is the $n$th partial sum of", + "$\\sum_{s=k}^\\infty a_s$, then", + "\\begin{eqnarray*}", + "T_r\\ar=b_1+b_2+\\cdots+b_r\\\\", + "\\ar=(a_1+\\cdots+a_{n_1})+(a_{n_1+1}+\\cdots+a_{n_2})+\\cdots+", + "(a_{n_{r-1}+1}+\\cdots+a_{n_r})\\\\", + "\\ar=A_{n_r}.", + "\\end{eqnarray*}", + "Thus, $\\{T_r\\}$ is a subsequence of $\\{A_n\\}$, so", + "$\\lim_{r\\to\\infty} T_r=\\lim_{n\\to\\infty}A_n=A$ by", + "Theorem~\\ref{thmtype:4.2.2}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.2.2" + ], + "ref_ids": [ + 90 + ] + } + ], + "ref_ids": [] + }, + { + "id": 110, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.24", + "categories": [], + "title": "", + "contents": [ + "If $\\sum_{n=1}^\\infty b_n$ is a rearrangement of an absolutely", + "convergent series $\\sum_{n=1}^\\infty a_n,$ then $\\sum_{n=1}^\\infty", + "b_n$ also converges absolutely$,$ and to the same sum$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$$", + "\\overline{A}_n=|a_1|+|a_2|+\\cdots+|a_n|\\mbox{\\quad and\\quad}", + "\\overline{B}_n=|b_1|+|b_2|+\\cdots+|b_n|.", + "$$", + "For each $n\\ge1$, there is an integer $k_n$ such that", + "$b_1$, $b_2$, \\dots, $b_n$ are included among", + "$a_1$, $a_2$, \\dots, $a_{k_n}$,", + "so $\\overline{B}_n\\le\\overline{A}_{k_n}$. Since", + "$\\{\\overline{A}_n\\}$ is bounded, so is $\\{\\overline{B}_n\\}$, and", + "therefore $\\sum |b_n|<\\infty$ (Theorem~\\ref{thmtype:4.3.8}).", + "Now let", + "\\begin{eqnarray*}", + "A_n\\ar=a_1+a_2+\\cdots+a_n,\\quad B_n=b_1+b_2+\\cdots+", + "b_n,\\\\", + "A\\ar=\\sum_{n=1}^\\infty a_n,\\mbox{\\quad and\\quad} B=\\sum_{n=1}^\\infty", + "b_n.", + "\\end{eqnarray*}", + "\\newpage", + "\\noindent", + "We must show that $A=B$. Suppose that $\\epsilon>0$. From Cauchy's", + "convergence criterion for series and the", + "absolute convergence of $\\sum a_n$, there is an", + "integer $N$ such that", + "\\vspace*{2pt}", + "$$", + "|a_{N+1}|+|a_{N+2}|+\\cdots+|a_{N+k}|<\\epsilon,\\quad k\\ge1.", + "$$", + "\\vspace*{2pt}", + "\\noindent\\hskip-.3em Choose $N_1$ so that $a_1$, $a_2$, \\dots, $a_N$", + "are included", + "among", + "$b_1$, $b_2$, \\dots, $b_{N_1}$. If $n\\ge N_1$, then $A_n$ and $B_n$", + "both", + "include the terms $a_1$, $a_2$, \\dots, $a_N$, which cancel on", + "subtraction;", + "thus, $|A_n-B_n|$ is dominated by the sum of the absolute values of", + "finitely many terms from $\\sum a_n$ with subscripts greater than $N$.", + "Since every such sum is less than~$\\epsilon$,", + "\\vspace*{2pt}", + "$$", + "|A_n-B_n|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N_1.", + "$$", + "\\vspace*{2pt}", + "Therefore, $\\lim_{n\\to\\infty}(A_n-B_n)=0$ and $A=B$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.8" + ], + "ref_ids": [ + 99 + ] + } + ], + "ref_ids": [] + }, + { + "id": 111, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.25", + "categories": [], + "title": "", + "contents": [ + "If $P=\\{a_{n_i}\\}_1^\\infty$ and", + "$Q=", + "\\{a_{m_j}\\}_1^\\infty$ are respectively the subsequences of all", + "positive and", + "negative terms in a conditionally convergent series $\\sum a_n,$ then", + "\\begin{equation} \\label{eq:4.3.24}", + "\\sum_{i=1}^\\infty a_{n_i}=\\infty\\mbox{\\quad and\\quad}\\sum_{j=1}^\\infty", + "a_{m_j}=-\\infty.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If both series in \\eqref{eq:4.3.24} converge, then $\\sum", + "a_n$ converges absolutely, while if one converges and the other", + "diverges, then $\\sum a_n$ diverges to $\\infty$ or $-\\infty$. Hence,", + "both must diverge." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 112, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.26", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\sum_{n=1}^\\infty a_n$ is conditionally convergent and", + " $\\mu$ and $\\nu$ are arbitrarily given in the extended", + "reals$,$ with $\\mu\\le\\nu.$ Then", + "the terms of $\\sum_{n=1}^\\infty a_n$", + "can be rearranged to form a series $\\sum_{n=1}^\\infty b_n$", + "with partial sums", + "$$", + "B_n=b_1+b_2+\\cdots+b_n,\\quad n\\ge1,", + "$$", + "such that", + "\\begin{equation}\\label{eq:4.3.25}", + "\\limsup_{n\\to\\infty}B_n=\\nu\\mbox{\\quad and\\quad}", + "\\liminf_{n\\to\\infty}B_n=\\mu.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We consider the case where $\\mu$ and $\\nu$ are finite and leave", + "the other cases to you (Exercise~\\ref{exer:4.3.36}).", + "We may ignore any zero terms that occur in $\\sum_{n=1}^\\infty a_n$.", + "For convenience, we", + "denote the positive terms by", + " $P=\\{\\alpha_i\\}_1^\\infty$ and and the negative terms by", + "$Q=\\{-\\beta_j\\}_1^\\infty$. We construct the sequence", + "\\begin{equation} \\label{eq:4.3.26}", + "\\{b_n\\}_1^\\infty=\\{\\alpha_1, \\dots,\\alpha_{m_1},-\\beta_1, \\dots,-\\beta_{n_1},", + "\\alpha_{m_1+1}, \\dots,\\alpha_{m_2},-\\beta_{n_1+1}, \\dots,-\\beta_{n_2},", + "\\dots\\},", + "\\end{equation}", + "\\newpage", + "\\noindent", + "with segments chosen alternately from $P$ and $Q$. Let $m_0=n_0=0$.", + "If $k\\ge1$, let $m_k$ and $n_k$ be the smallest integers such that", + "$m_k>m_{k-1}$, $n_k>n_{k-1}$,", + "$$", + "\\sum_{i=1}^{m_k}\\alpha_i-\\sum_{j=1}^{n_{k-1}}\\beta_j\\ge\\nu,", + "\\mbox{\\quad and \\quad}", + "\\sum_{i=1}^{m_k}\\alpha_i-\\sum_{j=1}^{n_k}\\beta_j\\le\\mu.", + "$$", + "Theorem~\\ref{thmtype:4.3.25} implies", + "that this construction is possible:", + "since $\\sum \\alpha_i=\\sum\\beta_j=\\infty$, we", + "can choose $m_k$ and $n_k$ so that", + "$$", + "\\sum_{i=m_{k-1}}^{m_k}\\alpha_i\\mbox{\\quad and\\quad}", + "\\sum_{j=n_{k-1}}^{n_k}\\beta_j", + "$$", + "are as large as we please, no matter how large $m_{k-1}$ and $n_{k-1}$", + "are (Exercise~\\ref{exer:4.3.23}).", + "Since $m_k$ and $n_k$ are the smallest integers with the specified", + "properties,", + "\\begin{eqnarray}", + "\\nu\\le B_{m_k+n_{k-1}}\\ar<\\nu+\\alpha_{m_k},\\quad k\\ge2,", + "\\label{eq:4.3.27}\\\\", + "\\arraytext{and}\\nonumber\\\\", + "\\mu-\\beta_{n_k}\\ar0$ if $m_k+n_k< n\\le m_{k+1}+n_k$, so", + "\\begin{equation}\\label{eq:4.3.30}", + "B_{m_k+n_k}\\le B_n\\le B_{m_{k+1}+n_k},\\quad m_k+n_k\\le n\\le m_{k+1}+n_k.", + "\\end{equation}", + "Because of \\eqref{eq:4.3.27} and \\eqref{eq:4.3.28}, \\eqref{eq:4.3.29}", + "and \\eqref{eq:4.3.30} imply that", + "\\begin{eqnarray}", + "\\mu-\\beta_{n_k}\\ar0$ then", + " $B_n>\\nu+ \\epsilon$ for only finitely many", + "values of $n$. Therefore,", + "$\\limsup_{n\\to\\infty} B_n=\\nu$.", + "From the second inequality in \\eqref{eq:4.3.28}, $B_n\\le \\mu$ for", + "infinitely many values of $n$. However, since", + "$\\lim_{j\\to\\infty}\\beta_j=0$,", + "the first inequalities in \\eqref{eq:4.3.31} and \\eqref{eq:4.3.32}", + "imply that if $\\epsilon>0$ then", + " $B_n<\\mu-\\epsilon$ for only finitely many", + "values of $n$. Therefore,", + "$\\liminf_{n\\to\\infty} B_n=\\mu$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.25" + ], + "ref_ids": [ + 111 + ] + } + ], + "ref_ids": [] + }, + { + "id": 113, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.27", + "categories": [], + "title": "", + "contents": [ + "Let", + "$$", + "\\sum_{n=0}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=0}^\\infty b_n=B,", + "$$", + "where $A$ and $B$ are finite, and at least one term of each series", + "is nonzero. Then $\\sum_{n=0}^\\infty p_n=AB$ for every sequence", + "$\\{p_n\\}$ obtained by ordering the products in $\\eqref{eq:4.3.33}$ if and", + "only if $\\sum a_n$ and $\\sum b_n$ converge absolutely$.$ Moreover$,$", + "in this case, $\\sum p_n$ converges absolutely$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "First, let $\\{p_n\\}$ be the sequence obtained by", + "arranging the products $\\{a_ib_j\\}$ according to the scheme indicated in", + "\\eqref{eq:4.3.34}, and define", + "$$", + "\\begin{array}{ll}", + "A_n=a_0+a_1+\\cdots+a_n,&", + "\\overline{A}_n=|a_0|+|a_1|+\\cdots+|a_n|,\\\\[2\\jot]", + "B_n=b_0+b_1+\\cdots+b_n,&", + "\\overline{B}_n=|b_0|+|b_1|+\\cdots+|b_n|,\\\\[2\\jot]", + "P_n\\hskip.1em=p_0+p_1+\\cdots+p_n,&\\overline{P}_n\\hskip.1em=|p_0|+|p_1|+\\cdots+|p_n|.", + "\\end{array}", + "$$", + "From \\eqref{eq:4.3.34}, we see that", + "$$", + "P_0=A_0B_0,\\quad P_3=A_1B_1,\\quad P_8=A_2B_2,", + "$$", + "and, in general,", + "\\begin{equation}\\label{eq:4.3.36}", + "P_{(m+1)^2-1}=A_mB_m.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Similarly,", + "\\begin{equation}\\label{eq:4.3.37}", + "\\overline{P}_{(m+1)^2-1}=\\overline{A}_m\\overline{B}_m.", + "\\end{equation}", + "If $\\sum |a_n|<\\infty$ and $\\sum |b_n|<\\infty$, then", + "$\\{\\overline{A}_m\\overline{B}_m\\}$ is bounded and, since", + "$\\overline{P}_m\\le\\overline{P}_{(m+1)^2-1}$,", + "\\eqref{eq:4.3.37} implies that $\\{\\overline{P}_m\\}$ is bounded. Therefore,", + "$\\sum |p_n| <\\infty$, so $\\sum p_n$ converges. Now", + "$$", + "\\begin{array}{rcll}", + "\\dst{\\sum ^\\infty_{n=0}p_n}\\ar=\\dst{\\lim_{n\\to\\infty}P_n}&\\mbox{(by", + "definition)}\\\\[2\\jot]", + "\\ar=\\dst{\\lim_{m\\to\\infty} P_{(m+1)^2-1}}&\\mbox{(by", + "Theorem~\\ref{thmtype:4.2.2})}\\\\[2\\jot]", + "\\ar=\\dst{\\lim_{m\\to\\infty} A_mB_m}&\\mbox{(from \\eqref{eq:4.3.36})}\\\\[2\\jot]", + "\\ar=\\dst{\\left(\\lim_{m\\to\\infty}", + "A_m\\right)\\left(\\lim_{m\\to\\infty}B_m\\right)}", + "&\\mbox{(by Theorem~\\ref{thmtype:4.1.8})}\\\\[2\\jot]", + "\\ar=AB.", + "\\end{array}", + "$$", + "Since any other ordering of the products in \\eqref{eq:4.3.33} produces a", + " a rearrangement of the", + "absolutely convergent series $\\sum_{n=0}^\\infty p_n$,", + "Theorem~\\ref{thmtype:4.3.24} implies that $\\sum |q_n|<\\infty$ for every", + "such ordering and that $\\sum_{n=0}^\\infty q_n=AB$. This shows that", + "the stated condition is sufficient.", + "For necessity, again let $\\sum_{n=0}^\\infty p_n$ be obtained from the", + "ordering indicated in \\eqref{eq:4.3.34}, and suppose that $\\sum_{n=0}^\\infty p_n$ and all its", + "rearrangements converge to $AB$. Then $\\sum p_n$ must converge", + "absolutely, by Theorem~\\ref{thmtype:4.3.26}. Therefore,", + "$\\{\\overline{P}_{m^2-1}\\}$ is bounded, and \\eqref{eq:4.3.37} implies that", + "$\\{\\overline{A}_m\\}$ and $\\{\\overline{B}_m\\}$ are bounded.", + "(Here we need", + "the assumption that neither $\\sum a_n$ nor $\\sum b_n$ consists", + "entirely of zeros. Why?)", + " Therefore,", + "$\\sum |a_n|<\\infty$ and $\\sum |b_n|<\\infty$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.2.2", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.8", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.24", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.26" + ], + "ref_ids": [ + 90, + 85, + 110, + 112 + ] + } + ], + "ref_ids": [] + }, + { + "id": 114, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.29", + "categories": [], + "title": "", + "contents": [ + "If $\\sum_{n=0}^\\infty a_n$ and", + "$\\sum_{n=0}^\\infty b_n$ converge absolutely to sums $A$ and $B,$ then", + "the Cauchy product of $\\sum_{n=0}^\\infty a_n$", + "and $\\sum_{n=0}^\\infty b_n$", + "converges absolutely to $AB.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $C_n$ be the $n$th partial sum of the Cauchy", + "product; that is,", + "$$", + "C_n=c_0+c_1+\\cdots+c_n", + "$$", + "(see \\eqref{eq:4.3.38}). Let $\\sum_{n=0}^\\infty p_n$ be the series", + "obtained", + "by ordering the products $\\{a_i,b_j\\}$ according to the scheme", + "indicated in \\eqref{eq:4.3.35}, and define $P_n$ to be its $n$th partial", + "sum; thus,", + "$$", + "P_n=p_0+p_1+\\cdots+p_n.", + "$$", + "Inspection of \\eqref{eq:4.3.35} shows that $c_n$ is the sum of the $n+1$", + "terms connected by the diagonal arrows. Therefore, $C_n=P_{m_n}$,", + "where", + "$$", + "m_n=1+2+\\cdots+(n+1)-1=\\frac{n(n+3)}{2}.", + "$$", + "From Theorem~\\ref{thmtype:4.3.27}, $\\lim_{n\\to\\infty} P_{m_n}=AB$, so", + "$\\lim_{n\\to\\infty} C_n=AB$. To see that $\\sum |c_n|<\\infty$, we", + "observe that", + "$$", + "\\sum_{r=0}^n |c_r|\\le\\sum_{s=0}^{m_n} |p_s|", + "$$", + "\\nopagebreak", + "and recall that $\\sum |p_s|<\\infty$, from Theorem~\\ref{thmtype:4.3.27}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.27", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.27" + ], + "ref_ids": [ + 113, + 113 + ] + } + ], + "ref_ids": [] + }, + { + "id": 115, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.4", + "categories": [], + "title": "", + "contents": [ + "Let $\\{F_n\\}$ be defined on $S.$", + "Then", + "\\begin{alist}", + "\\item % (a)", + "$\\{F_n\\}$ converges pointwise to $F$ on $S$ if and only if there is,", + "for each $\\epsilon>0$ and $x\\in S$, an integer $N$ $($which may depend", + "on $x$ as well as $\\epsilon)$ such that", + "$$", + "|F_n(x)-F(x)|<\\epsilon\\mbox{\\quad if\\quad}\\ n\\ge N.", + "$$", + "\\item % (b)", + " $\\{F_n\\}$ converges uniformly to $F$ on $S$ if and only if", + "there is for each $\\epsilon>0$ an integer $N$ $($which depends only on", + "$\\epsilon$ and not on any particular $x$ in $S)$ such that", + "$$", + "|F_n(x)-F(x)|<\\epsilon\\mbox{\\quad for all $x$ in $S$ if $n\\ge N$}.", + "$$", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 116, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.5", + "categories": [], + "title": "", + "contents": [ + "If $\\{F_n\\}$ converges uniformly to $F$ on $S,$ then $\\{F_n\\}$ converges", + "pointwise to $F$ on $S.$ The converse is false$;$ that is$,$ pointwise", + "convergence does not imply uniform convergence." + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 117, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.6", + "categories": [], + "title": "Cauchy's Uniform Convergence Criterion", + "contents": [ + "A sequence of functions $\\{F_n\\}$ converges uniformly on a set $S$ if", + "and", + "only if for each $\\epsilon>0$ there is an integer $N$ such that", + "\\begin{equation} \\label{eq:4.4.2}", + "\\|F_n-F_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For necessity, suppose that $\\{F_n\\}$ converges uniformly to", + "$F$ on $S$. Then, if $\\epsilon>0$, there is an integer $N$ such that", + "$$", + "\\|F_k-F\\|_S<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} k\\ge N.", + "$$", + "Therefore,", + "\\begin{eqnarray*}", + "\\|F_n-F_m\\|_S\\ar=\\|(F_n-F)+(F-F_m)\\|_S\\\\", + "\\ar\\le \\|F_n-F\\|_S+\\|F-F_m\\|_S \\mbox{\\quad", + "(Lemma~\\ref{thmtype:4.4.2})\\quad}\\\\", + "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\mbox{\\quad if\\quad}", + "m, n\\ge N.", + "\\end{eqnarray*}", + "For sufficiency, we first observe that \\eqref{eq:4.4.2} implies that", + "$$", + "|F_n(x)-F_m(x)|<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N,", + "$$", + "for any fixed $x$ in $S$. Therefore, Cauchy's convergence criterion", + "for sequences of constants (Theorem~\\ref{thmtype:4.1.13})", + "implies that", + "$\\{F_n(x)\\}$ converges for each $x$ in $S$; that is, $\\{F_n\\}$", + "converges pointwise to a limit function $F$ on $S$. To see that the", + "convergence is uniform, we write", + "\\begin{eqnarray*}", + "|F_m(x)-F(x) |\\ar=|[F_m(x)-F_n(x)]+[F_n(x)-F(x)]|\\\\", + "\\ar\\le |F_m(x)-F_n(x)|+| F_n(x)-F(x)|\\\\", + "\\ar\\le \\|F_m-F_n\\|_S+|F_n(x)-F(x)|.", + "\\end{eqnarray*}", + "This and \\eqref{eq:4.4.2} imply that", + "\\begin{equation} \\label{eq:4.4.3}", + "|F_m(x)-F(x)|<\\epsilon+|F_n(x)-F(x)|\\quad\\mbox {if}\\quad n, m\\ge N.", + "\\end{equation}", + "Since $\\lim_{n\\to\\infty}F_n(x)=F(x)$,", + "$$", + "|F_n(x)-F(x)|<\\epsilon", + "$$", + "for some $n\\ge N$, so \\eqref{eq:4.4.3} implies that", + "$$", + "|F_m(x)-F(x)|<2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", + "$$", + "But this inequality holds for all $x$ in $S$, so", + "$$", + "\\|F_m-F\\|_S\\le2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", + "$$", + "Since $\\epsilon$ is an arbitrary positive number, this implies that", + "$\\{F_n\\}$ converges uniformly to $F$ on~$S$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" + ], + "ref_ids": [ + 251, + 89 + ] + } + ], + "ref_ids": [] + }, + { + "id": 118, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.7", + "categories": [], + "title": "", + "contents": [ + "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is", + "continuous at a point $x_0$ in $S,$ then so is $F$. Similar", + "statements hold for continuity from the right and left$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that each $F_n$ is continuous at $x_0$.", + "If $x\\in S$ and $n\\ge1$, then", + "\\begin{equation} \\label{eq:4.4.8}", + "\\begin{array}{rcl}", + "|F(x)-F(x_0)|\\ar\\le |F(x)-F_n(x)|+|F_n(x)-F_n(x_0)|+|F_n(x_0)-F(x_0)|", + "\\\\", + "\\ar\\le |F_n(x)-F_n(x_0)|+2\\|F_n-F\\|_S.", + "\\end{array}", + "\\end{equation}", + "Suppose that $\\epsilon>0$. Since $\\{F_n\\}$ converges uniformly to $F$", + "on $S$, we can choose $n$ so that $\\|F_n-F\\|_S<\\epsilon$. For this", + "fixed $n$, \\eqref{eq:4.4.8} implies that", + "\\begin{equation} \\label{eq:4.4.9}", + "|F(x)-F(x_0)|<|F_n(x)-F_n(x_0)|+2\\epsilon,\\quad x\\in S.", + "\\end{equation}", + "Since $F_n$ is continuous at $x_0$, there is a $\\delta>0$ such that", + "$$", + "|F_n(x)-F_n(x_0)|<\\epsilon\\mbox{\\quad if\\quad} |x-x_0|<\\delta,", + "$$", + "so, from \\eqref{eq:4.4.9},", + "$$", + "|F(x)-F(x_0)|<3\\epsilon,\\mbox{\\quad if\\quad} |x-x_0|<\\delta.", + "$$", + "Therefore, $F$ is continuous at $x_0$. Similar", + "arguments apply to the assertions on", + "continuity from the right and left." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 119, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.9", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\{F_n\\}$ converges uniformly to $F$ on $S=[a,b]$. Assume", + "that $F$ and all $F_n$", + "are integrable on $[a,b].$ Then", + "\\begin{equation} \\label{eq:4.4.10}", + "\\int_a^b F(x)\\,dx=\\lim_{n\\to\\infty}\\int_a^b F_n(x)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since", + "\\begin{eqnarray*}", + "\\left|\\int_a^b F_n(x)\\,dx-\\int_a^b F(x)\\,dx\\right|\\ar\\le \\int_a^b", + "|F_n(x)-F(x)|\\,dx\\\\", + "\\ar\\le (b-a)\\|F_n-F\\|_S", + "\\end{eqnarray*}", + "and $\\lim_{n\\to\\infty}\\|F_n-F\\|_S=0$, the conclusion follows." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 120, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.10", + "categories": [], + "title": "", + "contents": [ + " Suppose that $\\{F_n\\}$ converges", + "pointwise to $F$ and each $F_n$ is integrable on $[a,b].$", + "\\begin{alist}", + "\\item % (a)", + "If the convergence is uniform$,$ then $F$ is integrable on", + "$[a,b]$ and $\\eqref{eq:4.4.10}$ holds.", + "\\item % (b)", + "If the sequence $\\{\\|F_n\\|_{[a,b]}\\}$ is bounded and $F$ is", + "integrable on $[a,b],$ then $\\eqref{eq:4.4.10}$ holds.", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 121, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.11", + "categories": [], + "title": "", + "contents": [ + "Suppose that $F'_n$ is continuous on $[a,b]$ for all $n\\ge1$ and $\\{F'_n\\}$", + "converges uniformly on $[a,b].$ Suppose also that", + " $\\{F_n(x_0)\\}$ converges for some $x_0$ in $[a,b].$ Then", + "$\\{F_n\\}$ converges uniformly on $[a,b]$ to a differentiable limit", + "function $F,$ and", + "\\begin{equation} \\label{eq:4.4.11}", + "F'(x)=\\lim_{n\\to\\infty}F'_n(x),\\quad a0$ there is an integer $N$ such that", + "\\vskip0pt", + "\\begin{equation} \\label{eq:4.4.16}", + "\\|f_n+f_{n+1}+\\cdots+f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge", + "N.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Apply Theorem~\\ref{thmtype:4.4.6} to the partial sums of", + "$\\sum f_n$, observing that", + "$$", + "f_n+f_{n+1}+\\cdots+f_m=F_m-F_{n-1}.", + "$$", + "\\vskip-2em" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.6" + ], + "ref_ids": [ + 117 + ] + } + ], + "ref_ids": [] + }, + { + "id": 123, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", + "categories": [], + "title": "Weierstrass's Test", + "contents": [ + "The series $\\sum f_n$ converges uniformly on $S$ if", + "\\begin{equation} \\label{eq:4.4.17}", + "\\|f_n\\|_S\\le M_n,\\quad n\\ge k,", + "\\end{equation}", + "where $\\sum M_n<\\infty.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Cauchy's convergence criterion for series of constants,", + "there is for each $\\epsilon>0$ an integer $N$ such that", + "$$", + "M_n+M_{n+1}+\\cdots+M_m<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N,", + "$$", + "which, because of \\eqref{eq:4.4.17}, implies that", + "$$", + "\\|f_n\\|_S+\\|f_{n+1}\\|_S+\\cdots+\\|f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad}", + " m, n\\ge N.", + "$$", + " Lemma~\\ref{thmtype:4.4.2} and Theorem~\\ref{thmtype:4.4.13} imply that", + "$\\sum f_n$ converges uniformly on $S$.", + "\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", + "TRENCH_REAL_ANALYSIS-thmtype:4.4.13" + ], + "ref_ids": [ + 251, + 122 + ] + } + ], + "ref_ids": [] + }, + { + "id": 124, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.16", + "categories": [], + "title": "Dirichlet's Test for Uniform Convergence", + "contents": [ + "The series", + "$$", + "\\sum_{n=k}^\\infty f_ng_n", + "$$", + " converges uniformly on", + "$S$ if", + " $\\{f_n\\}$ converges uniformly to zero on $S,$", + " $\\sum (f_{n+1}-f_n)$ converges absolutely uniformly on", + "$S,$ and", + "\\begin{equation} \\label{eq:4.4.19}", + "\\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k,", + "\\end{equation}", + "for some constant $M.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The proof is similar to the proof of", + "Theorem~\\ref{thmtype:4.3.20}. Let", + "$$", + "G_n=g_k+g_{k+1}+\\cdots+g_n,", + "$$", + "and consider the partial sums of $\\sum_{n=k}^\\infty f_ng_n$:", + "\\begin{equation} \\label{eq:4.4.20}", + "H_n=f_kg_k+f_{k+1}g_{k+1}+\\cdots+f_ng_n.", + "\\end{equation}", + "By substituting", + "$$", + "g_k=G_k\\mbox{\\quad and\\quad} g_n=G_n-G_{n-1},\\quad n\\ge k+1,", + "$$", + "into \\eqref{eq:4.4.20}, we obtain", + "$$", + "H_n=f_kG_k+f_{k+1}(G_{k+1}-G_k)+\\cdots+f_n(G_n-G_{n-1}),", + "$$", + "which we rewrite as", + "$$", + "H_n=(f_k-f_{k+1})", + "G_k+(f_{k+1}-f_{k+2})G_{k+1}+\\cdots+(f_{n-1}-f_n)G_{n-1}+f_nG_n,", + "$$", + "or", + "\\begin{equation} \\label{eq:4.4.21}", + "H_n=J_{n-1}+f_nG_n,", + "\\end{equation}", + "where", + "\\begin{equation} \\label{eq:4.4.22}", + "J_{n-1}=(f_k-f_{k+1})G_k+(f_{k+1}-f_{k+2})", + "G_{k+1}+\\cdots+(f_{n-1}-f_n)G_{n-1}.", + "\\end{equation}", + "That is, $\\{J_n\\}$ is the sequence of partial sums of the series", + "\\begin{equation} \\label{eq:4.4.23}", + "\\sum_{j=k}^\\infty (f_j-f_{j+1})G_j.", + "\\end{equation}", + " From \\eqref{eq:4.4.19} and the definition of", + "$G_j$,", + "$$", + "\\left|\\sum^m_{j=n}[f_j(x)-f_{j+1}(x)]G_j(x)\\right|\\le M", + "\\sum^m_{j=n}|f_j(x)-f_{j+1}(x)|,\\quad x\\in S,", + "$$", + "\\newpage", + "\\noindent so", + "$$", + "\\left\\|\\sum^m_{j=n} (f_j-f_{j+1})G_j\\right\\|_S\\le M\\left\\|\\sum^m_{j=n}", + "|f_j-f_{j+1}|\\right\\|_S.", + "$$", + "Now suppose that $\\epsilon>0$.", + "Since $\\sum (f_j-f_{j+1})$ converges absolutely uniformly on $S$,", + "Theorem~\\ref{thmtype:4.4.13} implies that", + "there is an integer $N$ such that", + "the right side of the last", + "inequality is less than $\\epsilon$ if", + "$m\\ge n\\ge N$. The same is then true of the left side, so", + "Theorem~\\ref{thmtype:4.4.13}", + " implies that", + "\\eqref{eq:4.4.23} converges uniformly on~$S$.", + "We have now shown that $\\{J_n\\}$ as defined in \\eqref{eq:4.4.22} converges", + "uniformly to a limit function $J$ on $S$. Returning to \\eqref{eq:4.4.21},", + "we see that", + "$$", + "H_n-J=J_{n-1}-J+f_nG_n.", + "$$", + "Hence, from Lemma~\\ref{thmtype:4.4.2} and \\eqref{eq:4.4.19},", + "\\begin{eqnarray*}", + "\\|H_n-J\\|_S\\ar\\le \\|J_{n-1}-J\\|_S+\\|f_n\\|_S\\|G_n\\|_S\\\\", + "\\ar\\le \\|J_{n-1}-J\\|_S+M\\|f_n\\|_S.", + "\\end{eqnarray*}", + "Since $\\{J_{n-1}-J\\}$ and $\\{f_n\\}$ converge uniformly to zero on $S$,", + "it now follows that $\\lim_{n\\to\\infty}\\|H_n-J\\|_S=0$. Therefore,", + " $\\{H_n\\}$ converges uniformly on~$S$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.3.20", + "TRENCH_REAL_ANALYSIS-thmtype:4.4.13", + "TRENCH_REAL_ANALYSIS-thmtype:4.4.13", + "TRENCH_REAL_ANALYSIS-thmtype:4.4.2" + ], + "ref_ids": [ + 108, + 122, + 122, + 251 + ] + } + ], + "ref_ids": [] + }, + { + "id": 125, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.18", + "categories": [], + "title": "", + "contents": [ + "If $\\sum_{n=k}^\\infty f_n$ converges uniformly to $F$ on $S$ and each", + "$f_n$ is continuous at a point $x_0$ in $S,$ then so is $F.$ Similar", + "statements hold for continuity from the right and left$.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 126, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.19", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\sum_{n=k}^\\infty f_n$ converges uniformly to $F$ on", + "$S=[a,b].$ Assume that $F$ and $f_n,$ $n\\ge k,$", + "are integrable on $[a,b].$ Then", + "$$", + "\\int_a^b F(x)\\,dx=\\sum_{n=k}^\\infty \\int_a^b f_n(x)\\,dx.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 127, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.20", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f_n$ is continuously differentiable on $[a,b]$ for each", + "$n\\ge k,$ $\\sum_{n=k}^\\infty f_n(x_0)$ converges for some $x_0$ in", + "$[a,b],$ and", + "$\\sum_{n=k}^\\infty f'_n$ converges uniformly on $[a,b].$ Then", + "$\\sum_{n=k}^\\infty f_n$ converges uniformly on $[a,b]$ to a", + "differentiable function $F,$ and", + "$$", + "F'(x)=\\sum_{n=k}^\\infty f'_n(x),\\quad aR.$ No general statement can be made concerning convergence", + "at the endpoints $x=x_0+R$ and $x=x_0-R:$ the series may converge", + "absolutely or conditionally at both$,$ converge conditionally at one", + "and diverge at the other$,$ or diverge at both$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "In any case, the series \\eqref{eq:4.5.1} converges to $a_0$ if", + "$x=x_0$. If", + "\\begin{equation}\\label{eq:4.5.3}", + "\\sum |a_n|r^n<\\infty", + "\\end{equation}", + "for some $r>0$, then $\\sum a_n (x-x_0)^n$ converges", + "absolutely uniformly in $[x_0-r, x_0+r]$, by Weierstrass's test", + "(Theorem~\\ref{thmtype:4.4.15}) and", + "Exercise~\\ref{exer:4.4.21}. From Cauchy's root test", + "(Theorem~\\ref{thmtype:4.3.17}),", + "\\eqref{eq:4.5.3} holds if", + "$$", + "\\limsup_{n\\to\\infty} (|a_n|r^n)^{1/n}<1,", + "$$", + "which is equivalent to", + " $$", + " r\\,\\limsup_{n\\to\\infty} |a_n|^{1/n}<1", + "$$", + "(Exercise~\\ref{exer:4.1.30}\\part{a}).", + "From \\eqref{eq:4.5.2}, this can be rewritten as $rR$, then", + "\\newpage", + "$$", + "\\frac{1}{ R}>\\frac{1}{ |x-x_0|},", + "$$", + "so \\eqref{eq:4.5.2} implies that", + "$$", + "|a_n|^{1/n}\\ge\\frac{1}{ |x-x_0|}\\mbox{\\quad and therefore\\quad}", + "|a_n(x-x_0)^n|\\ge1", + "$$", + "for infinitely many values of $n$. Therefore, $\\sum a_n(x-x_0)^n$", + "diverges (Corollary~\\ref{thmtype:4.3.6}) if $|x-x_0|>R$.", + "In particular, the series diverges for all $x\\ne x_0$ if $R=0$.", + "To prove the assertions concerning the possibilities at $x=x_0+R$", + "and $x=x_0-R$ requires examples, which follow. (Also, see", + "Exercise~\\ref{exer:4.5.1}.)" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" + ], + "ref_ids": [ + 123, + 106, + 277 + ] + } + ], + "ref_ids": [] + }, + { + "id": 129, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.3", + "categories": [], + "title": "", + "contents": [ + "The radius of convergence of $\\sum", + "a_n(x-x_0)^n$ is given by", + "$$", + "\\frac{1}{ R}=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|", + "$$", + "if the limit exists in the extended reals$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Theorem~\\ref{thmtype:4.5.2}, it suffices to show that if", + "\\begin{equation}\\label{eq:4.5.4}", + "L=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right|", + "\\end{equation}", + "exists in the extended reals, then", + "\\begin{equation}\\label{eq:4.5.5}", + "L=\\limsup_{n\\to\\infty}|a_n|^{1/n}.", + "\\end{equation}", + "We will show that this is so if $0 N.", + "$$", + "Therefore, if", + "$$", + "K_1=|a_N|(L-\\epsilon)^{-N}\\mbox{\\quad and\\quad} K_2=|a_N|(L+", + "\\epsilon)^{-N},", + "$$", + "then", + "\\begin{equation}\\label{eq:4.5.6}", + "K^{1/n}_1(L-\\epsilon)<|a_n|^{1/n}0$, choose $N$ so that", + "$$", + "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N+1.", + "$$", + "Then, if $00$. If", + "$$", + "|\\mathbf{X}_1-\\mathbf{X}_0|0$ there is an integer $K$ such that", + "$$", + "|\\mathbf{X}_r-\\mathbf{X}_s|<\\epsilon\\mbox{\\quad if\\quad} r,s\\ge K.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 142, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.17", + "categories": [], + "title": "Principle of Nested Sets", + "contents": [ + "If $S_1,$ $S_2,$ \\dots\\ are closed nonempty subsets of $\\R^n$", + "such that", + "\\begin{equation}\\label{eq:5.1.14}", + "S_1\\supset S_2\\supset\\cdots\\supset S_r\\supset\\cdots", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:5.1.15}", + "\\lim_{r\\to\\infty} d(S_r)=0,", + "\\end{equation}", + "then the intersection", + "$$", + "I=\\bigcap^\\infty_{r=1}S_r", + "$$", + "contains exactly one point$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$\\{\\mathbf{X}_r\\}$ be a sequence such that $\\mathbf{X}_r\\in S_r\\ (r\\ge1)$.", + "Because of", + "\\eqref{eq:5.1.14}, $\\mathbf{X}_r\\in S_k$ if $r\\ge k$, so", + "$$", + "|\\mathbf{X}_r-\\mathbf{X}_s|2$. The counterpart of the", + "square $T$ is the {\\it hypercube\\/} with sides of", + "length", + "$L$:", + "$$", + "T=\\set{(x_1,x_2, \\dots,x_n)}{ a_i\\le x_i\\le a_i+L, i=1,2, \\dots, n}.", + "$$", + "Halving the intervals of variation of the $n$ coordinates", + "$x_1$, $x_2$, \\dots, $x_n$ divides $T$ into $2^n$ closed hypercubes", + "with sides of length $L/2$:", + "$$", + "T^{(i)}=\\set{(x_1,x_2, \\dots,x_n)}{b_i\\le x_i\\le b_i+L/2,", + "1\\le i\\le n},", + "$$", + "where $b_i=a_i$ or $b_i=a_i+L/2$. If no finite subcollection of ${\\mathcal", + "H}$ covers $S$, then at least one of these smaller hypercubes must", + "contain a subset of $S$ that is not covered by any finite subcollection", + "of $S$. Now the proof proceeds as for $n=2$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.1.17" + ], + "ref_ids": [ + 142 + ] + } + ], + "ref_ids": [] + }, + { + "id": 144, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.20", + "categories": [], + "title": "", + "contents": [ + " An open set $S$ in $\\R^n$ is", + "connected if and only if it is polygonally connected$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For sufficiency, we will show that if $S$ is disconnected, then", + "$S$ is not poly\\-gonally connected. Let $S=A\\cup B$, where $A$ and $B$", + "satisfy \\eqref{eq:5.1.16}. Suppose that $\\mathbf{X}_1\\in A$ and $\\mathbf{X}_2\\in", + "B$, and assume that there is a polygonal path in $S$ connecting", + "$\\mathbf{X}_{1}$ to $\\mathbf{X}_2$. Then some line segment $L$ in this", + "path must", + "contain a point $\\mathbf{Y}_1$ in $A$ and a point $\\mathbf{Y}_2$ in $B$. The", + "line segment", + "$$", + "\\mathbf{X}=t\\mathbf{Y}_2+(1-t)\\mathbf{Y}_1,\\quad 0\\le t\\le1,", + "$$", + "is part of $L$ and therefore in $S$. Now define", + "$$", + "\\rho=\\sup\\set{\\tau}{ tY_2+(1-t)\\mathbf{Y}_1\\in A,\\ 0\\le t\\le", + "\\tau\\le1},", + "$$", + "and let", + "$$", + "\\mathbf{X}_\\rho=\\rho\\mathbf{Y}_2+(1-\\rho)\\mathbf{Y}_1.", + "$$", + "Then $\\mathbf{X}_\\rho\\in\\overline{A}\\cap\\overline{B}$. However, since", + "$\\mathbf{X}_\\rho\\in A\\cup B $ and $\\overline{A}\\cap", + "B=A\\cap\\overline{B}=\\emptyset$, this is impossible. Therefore,", + "the assumption that there is a polygonal path in $S$", + "from $\\mathbf{X}_1$ to $\\mathbf{X}_2$ must be false.", + "For necessity, suppose that $S$ is a connected open set and $\\mathbf{X}_0\\in", + "S$. Let $A$ be the set consisting of $\\mathbf{X}_0$ and the points in $S$", + "can be connected to $\\mathbf{X}_0$ by polygonal paths in $S$. Let $B$ be", + "set of points in $S$ that cannot be connected to $\\mathbf{X}_0$", + "by polygonal paths.", + " If $\\mathbf{Y}_0\\in S$, then $S$ contains an", + "$\\epsilon$-neighborhood $N_\\epsilon (\\mathbf{Y}_0)$ of $\\mathbf{Y}_0$,", + "since $S$ is open. Any point $\\mathbf{Y}_1$ in $N_\\epsilon", + "(\\mathbf{Y}_{0}$", + " can be connected to $\\mathbf{Y}_0$ by the line segment", + "$$", + "\\mathbf{X}=t\\mathbf{Y}_1+(1-t)\\mathbf{Y}_0,\\quad 0\\le t\\le1,", + "$$", + "which lies in $N_\\epsilon(\\mathbf{Y}_0)$ (Lemma~\\ref{thmtype:5.1.12}) and", + "therefore in", + "$S$. This implies that $\\mathbf{Y}_0$ can be connected to $\\mathbf{X}_0$ by a", + "polygonal path in $S$ if and only if every member of $N_\\epsilon", + "(\\mathbf{Y}_{0})$", + " can also. Thus, $N_\\epsilon(\\mathbf{Y}_0)\\subset A$ if $\\mathbf{Y}_0\\in", + "A$, and $N_\\epsilon (\\mathbf{Y}_0)\\in B$ if $\\mathbf{Y}_0\\in B$. Therefore,", + "$A$ and $B$ are open. Since $A\\cap B =\\emptyset$, this implies that", + "$A\\cap\\overline{B}=\\overline{A}\\cap B=\\emptyset$", + "(Exercise~\\ref{exer:5.1.14}). Since $A$ is nonempty $(\\mathbf{X}_0\\in A)$,", + "it", + "now follows that $B=\\emptyset$, since if $B\\ne\\emptyset$, $S$ would be", + "disconnected (Definition~\\ref{thmtype:5.1.19}). Therefore, $A=S$, which", + "completes the proof of necessity." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.1.12", + "TRENCH_REAL_ANALYSIS-thmtype:5.1.19" + ], + "ref_ids": [ + 253, + 343 + ] + } + ], + "ref_ids": [] + }, + { + "id": 145, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.2", + "categories": [], + "title": "", + "contents": [ + " If $\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})$ exists$,$ then it is", + "unique." + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 146, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.3", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ and $g$ are defined on a set $D,$ $\\mathbf{X}_0$ is a", + "limit point of $D,$ and", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L_1,\\quad\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} g(\\mathbf{X})=L_2.", + "$$", + "Then", + "\\begin{eqnarray}", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f+g)(\\mathbf{X})\\ar=L_1+L_2,\\label{eq:5.2.10}\\\\", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f-g)(\\mathbf{X})\\ar=L_1-L_2,\\label{eq:5.2.11}\\\\", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(fg)(\\mathbf{X})\\ar=L_1L_2,\\label{eq:5.2.12}\\\\", + "\\arraytext{and$,$ if $L_2\\ne0,$}\\nonumber\\\\", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}\\left(\\frac{f}{ g}\\right)(\\mathbf{X})", + "\\ar=\\frac{L_1}{ L_2}.\\label{eq:5.2.13}", + "\\end{eqnarray}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 147, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.7", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f.$ Then", + "$f$", + "is continuous at $\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there", + "is a $\\delta>0$ such that", + "$$", + "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)\\right|<\\epsilon", + "$$", + "whenever", + "$$", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 148, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are continuous on a set $S$ in $\\R^n,$ then so", + "are $f+g,$ $f-g,$ and $fg.$ Also$,$ $f/g$ is continuous at each", + "$\\mathbf{X}_0$ in $S$ such that $g(\\mathbf{X}_0)\\ne0.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 149, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.9", + "categories": [], + "title": "", + "contents": [ + "For a vector-valued function $\\mathbf{G},$", + "$$", + "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{G}(\\mathbf{U})=\\mathbf{L}", + "$$", + "if and only if for each $\\epsilon>0$ there is a $\\delta>0$ such that", + "$$", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{L}|<\\epsilon\\mbox{\\quad whenever\\quad}", + "0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}.", + "$$", + "Similarly, $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$ if and only if for", + "each", + "$\\epsilon> 0$ there is a $\\delta>0$ such that", + "$$", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon", + "\\mbox{\\quad whenever\\quad}", + " |\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 150, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.10", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be a real-valued function defined on a subset of $\\R^n,$", + " and let the", + "vector-valued function $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ be defined on a", + "domain $D_\\mathbf{G}$ in $\\R^m.$ Let the set", + "$$", + "T=\\set{\\mathbf{U}}{\\mathbf{U}\\in D_{\\mathbf{G}}\\mbox{\\quad and \\quad}", + "\\mathbf{G}(\\mathbf{U})\\in D_f}", + "$$", + "$($Figure~\\ref{figure:5.2.3}$)$,", + " be", + "nonempty$,$ and define the real-valued composite function", + "$$", + "h=f\\circ\\mathbf{G}", + "$$", + "on $T$ by", + "$$", + "h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U})),\\quad \\mathbf{U}\\in T.", + "$$", + "Now suppose that $\\mathbf{U}_0$ is in $T$ and is a limit point of $T,$", + "$\\mathbf{G}$ is continuous at $\\mathbf{U}_0,$ and $f$ is continuous at", + "$\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then $h$ is continuous at", + "$\\mathbf{U}_0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $\\epsilon>0$. Since $f$ is continuous at", + "$\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0)$, there is an $\\epsilon_1>0$", + "such that", + "\\begin{equation}\\label{eq:5.2.17}", + "|f(\\mathbf{X})-f(\\mathbf{G}(\\mathbf{U}_0))|<\\epsilon", + "\\end{equation}", + "if", + "\\begin{equation}\\label{eq:5.2.18}", + "|\\mathbf{X}-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon_1\\mbox{\\quad and\\quad}", + "\\mathbf{X}\\in D_f.", + "\\end{equation}", + "Since $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$, there is a $\\delta>0$", + "such that", + "$$", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon_1", + "\\mbox{\\quad if\\quad} |\\mathbf{U}-\\mathbf{U}_0|<", + "\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_\\mathbf{G}.", + "$$", + "By taking $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ in \\eqref{eq:5.2.17} and", + "\\eqref{eq:5.2.18}, we see that", + "$$", + "|h(\\mathbf{U})-h(\\mathbf{U}_0)|=|f(\\mathbf{G}(\\mathbf{U})", + "-f(\\mathbf{G}(\\mathbf{U}_0))|<\\epsilon", + "$$", + "if", + "$$", + "|\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in T.", + "$$" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 151, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.11", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a compact set $S$ in $\\R^n,$ then $f$", + "is bounded on~$S.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 152, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be continuous on a compact set $S$ in $\\R^n$ and", + "$$", + "\\alpha=\\inf_{\\mathbf{X}\\in S}f(\\mathbf{X}),\\quad\\beta=", + "\\sup_{\\mathbf{X}\\in S}f(\\mathbf{X}).", + "$$", + "Then", + "$$", + "f(\\mathbf{X}_1)=\\alpha\\mbox{\\quad and\\quad} f(\\mathbf{X}_2)=\\beta", + "$$", + "for some $\\mathbf{X}_1$ and $\\mathbf{X}_2$ in $S.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 153, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.13", + "categories": [], + "title": "Intermediate Value Theorem", + "contents": [ + "Let $f$ be continuous on a region $S$ in $\\R^n.$ Suppose that", + "$\\mathbf{A}$ and $\\mathbf{B}$ are in $S$ and", + "$$", + "f(\\mathbf{A})u}.", + "\\end{eqnarray*}", + "If $\\mathbf{X}_0\\in R$, the continuity of $f$ implies that there is a", + "$\\delta>0$ such that $f(\\mathbf{X})0$. Choose $\\delta>0$ so that", + "the open square", + "\\newpage", + "$$", + "S_\\delta=\\set{(x,y)}{|x-x_0|<\\delta, |y-y_0|<\\delta}", + "$$", + "is in $N$ and", + "\\begin{equation}\\label{eq:5.3.6}", + "|f_{xy}(\\widehat{x},\\widehat{y})-f_{xy}(x_0,y_0)|<\\epsilon\\quad", + "\\mbox{\\quad if\\quad}(\\widehat{x},\\widehat{y})\\in S_\\delta.", + "\\end{equation}", + "This is possible because of the continuity of $f_{xy}$ at $(x_0,y_0)$.", + "The function", + "\\begin{equation}\\label{eq:5.3.7}", + "A(h,k)=f(x_0+h, y_0+k)-f(x_0+h,y_0)-f(x_0,y_0+k)+f(x_0,y_0)", + "\\end{equation}", + "is defined if $-\\delta0$. Our assumptions imply that there is", + "a $\\delta>0$ such that $f_{x_1}, f_{x_2}, \\dots, f_{x_n}$ are defined", + "in the $n$-ball", + "$$", + "S_\\delta (\\mathbf{X}_0)=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\delta}", + "$$", + "and", + "\\begin{equation}\\label{eq:5.3.24}", + "|f_{x_j}(\\mathbf{X})-f_{x_j}(\\mathbf{X}_0)|<\\epsilon\\mbox{\\quad if\\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta,\\quad 1\\le j\\le n.", + "\\end{equation}", + "Let $\\mathbf{X}=(x_1,x_, \\dots,x_n)$ be in $S_\\delta(\\mathbf{X}_0)$.", + "Define", + "$$", + "\\mathbf{X}_j=(x_1, \\dots,x_j, x_{j+1,0}, \\dots,x_{n0}),\\quad 1\\le j\\le n-1,", + "$$", + "and", + "$\\mathbf{X}_n=\\mathbf{X}$.", + "Thus, for $1\\le j\\le n$, $\\mathbf{X}_j$ differs from $\\mathbf{X}_{j-1}$", + " in the", + "$j$th component only, and the line segment from $\\mathbf{X}_{j-1}$ to", + "$\\mathbf{X}_j$ is in $S_\\delta (\\mathbf{X}_0)$.", + "Now write", + "\\begin{equation}\\label{eq:5.3.25}", + "f(\\mathbf{X})-f(\\mathbf{X}_0)=f(\\mathbf{X}_n)-f(\\mathbf{X}_0)=", + "\\sum^n_{j=1}\\,[f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})],", + "\\end{equation}", + "and consider the auxiliary functions", + "\\begin{equation}\\label{eq:5.3.26}", + "\\begin{array}{rcl}", + "g_1(t)\\ar=f(t,x_{20}, \\dots,x_{n0}),\\\\[2\\jot]", + "g_j(t)\\ar=f(x_1, \\dots,x_{j-1},t,x_{j+1,0}, \\dots,x_{n0}),\\quad 2\\le j\\le", + "n-1,\\\\[2\\jot]", + "g_n(t)\\ar=f(x_1, \\dots,x_{n-1},t),", + "\\end{array}", + "\\end{equation}", + "where, in each case, all variables except $t$ are temporarily regarded", + "as constants. Since", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g_j(x_j)-g_j(x_{j0}),", + "$$", + "the mean value theorem implies that", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g'_j(\\tau_j)(x_j-x_{j0}),", + "$$", + "\\newpage", + "\\noindent", + "where $\\tau_j$ is between $x_j$ and $x_{j0}$. From \\eqref{eq:5.3.26},", + "$$", + "g'_j(\\tau_j)=f_{x_j}(\\widehat{\\mathbf{X}}_j),", + "$$", + "where $\\widehat{\\mathbf{X}}_j$ is on the line segment from $\\mathbf{X}_{j-1}$ to", + "$\\mathbf{X}_j$. Therefore,", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=f_{x_j}(\\widehat{\\mathbf{X}}_j)(x_j-x_{j0}),", + "$$", + "and \\eqref{eq:5.3.25} implies that", + "\\begin{eqnarray*}", + "f(\\mathbf{X})-f(\\mathbf{X}_0)\\ar=\\sum^n_{j=1} f_{x_j} (\\widehat{\\mathbf{X}}_j)(x_j-x_{j0})\\\\", + "\\ar=\\sum^n_{j=1} f_{x_j}(\\mathbf{X}_0) (x_j-x_{j0})+\\sum^n_{j=1}", + "\\,[f_{x_j}(\\widehat{\\mathbf{X}}_j)-f_{x_j}(\\mathbf{X}_0)](x_j-x_{j0}).", + "\\end{eqnarray*}", + "From this and \\eqref{eq:5.3.24},", + "$$", + "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)-\\sum^n_{j=1}", + "f_{x_j}(\\mathbf{X}_{0})", + "(x_j-x_{j0})\\right|\\le", + "\\epsilon\\sum^n_{j=1} |x_j-x_{j0}|\\le n\\epsilon |\\mathbf{X}-\\mathbf{X}_0|,", + "$$", + "which implies that $f$ is differentiable at $\\mathbf{X}_0$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 162, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.11", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is defined in a neighborhood of $\\mathbf{X}_0$ in", + "$\\R^n$ and $f_{x_1}(\\mathbf{X}_0),$ $f_{x_2}(\\mathbf{X}_{0}),$", + " \\dots$,$ $f_{x_n}(\\mathbf{X}_{0})$", + " exist$.$ Let $\\mathbf{X}_0$ be a local extreme point of $f.$ Then", + "\\begin{equation}\\label{eq:5.3.42}", + "f_{x_i}(\\mathbf{X}_0)=0,\\quad 1\\le i\\le n.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$$", + "\\mathbf{E}_1=(1,0, \\dots,0),\\quad \\mathbf{E}_{2}", + "=(0,1,0, \\dots,0),\\dots,\\quad \\mathbf{E}_n=", + "(0,0, \\dots,1),", + "$$", + "and", + "$$", + "g_i(t)=f(\\mathbf{X}_0+t\\mathbf{E}_i),\\quad 1\\le i\\le n.", + "$$", + "Then $g_i$ is differentiable at $t=0$, with", + "$$", + "g'_i(0)=f_{x_i}(\\mathbf{X}_0)", + "$$", + "\\newpage", + "\\noindent", + "(Definition~\\ref{thmtype:5.3.1}). Since $\\mathbf{X}_0$ is a local extreme", + "point of $f$, $t_0=0$ is a local extreme point of $g_i$. Now", + "Theorem~\\ref{thmtype:2.3.7} implies that $g'_i(0)=0$, and this", + "implies \\eqref{eq:5.3.42}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.3.1", + "TRENCH_REAL_ANALYSIS-thmtype:2.3.7" + ], + "ref_ids": [ + 349, + 31 + ] + } + ], + "ref_ids": [] + }, + { + "id": 163, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.3", + "categories": [], + "title": "The Chain Rule", + "contents": [ + "Suppose that the real-valued function $f$ is differentiable at", + "$\\mathbf{X}_0$", + "in $\\R^n,$ the vector-valued function $\\mathbf{G}", + "=(g_1,g_2, \\dots,g_n)$ is differentiable at", + "$\\mathbf{U}_0$ in $\\R^m,$ and $\\mathbf{X}_{0}", + " = \\mathbf{G}(\\mathbf{U}_0).$ Then the real-valued composite function", + "$h=f\\circ\\mathbf{G}$ defined by", + "\\begin{equation} \\label{eq:5.4.3}", + "h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U}))", + "\\end{equation}", + "is differentiable at $\\mathbf{U}_0,$ and", + "\\begin{equation} \\label{eq:5.4.4}", + "d_{\\mathbf{U}_0}h=f_{x_1}(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_1+f_{x_2}", + "(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_2+\\cdots", + "+f_{x_n} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_n.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We leave it to you to show that $\\mathbf{U}_0$ is an interior point", + "of the domain of $h$ (Exercise~\\ref{exer:5.4.1}), so it is legitimate to", + "ask if $h$ is differentiable at $\\mathbf{U}_0$.", + "Let $\\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0})$. Note that", + "$$", + "x_{i0}=g_i(\\mathbf{U}_0),\\quad", + "1\\le i\\le n,", + "$$", + "by assumption.", + "Since $f$ is differentiable at $\\mathbf{X}_0$,", + "Lemma~\\ref{thmtype:5.3.8} implies that", + "\\begin{equation} \\label{eq:5.4.5}", + "f(\\mathbf{X})-f(\\mathbf{X}_0)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)", + "(x_i-x_{i0})+E(\\mathbf{X})|\\mathbf{X}-\\mathbf{X}_0|,", + "\\end{equation}", + "where", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}E(\\mathbf{X})=0.", + "$$", + "\\newpage", + "\\noindent", + " Substituting $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$", + " and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0)$ in \\eqref{eq:5.4.5} and recalling", + "\\eqref{eq:5.4.3} yields", + "\\begin{equation} \\label{eq:5.4.6}", + "h(\\mathbf{U})-h(\\mathbf{U}_0)=\\dst{\\sum_{i=1}^n}\\, f_{x_i}(\\mathbf{X}_0)", + "(g_i(\\mathbf{U})-g_i(\\mathbf{U}_0))", + "+E(\\mathbf{G}(\\mathbf{U}))", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|.", + "\\end{equation}", + "Substituting \\eqref{eq:5.4.1} into \\eqref{eq:5.4.6} yields", + "$$", + "\\begin{array}{rcl}", + "h(\\mathbf{U})-h(\\mathbf{U}_0)\\ar=\\dst{\\sum_{i=1}^n} f_{x_i}(\\mathbf{X}_0)", + "(d_{\\mathbf{U}_0}g_i) (\\mathbf{U}-\\mathbf{U}_0)", + "+\\dst{\\left(\\sum_{i=1}^n", + "f_{x_i}(\\mathbf{X}_0)E_i(\\mathbf{U})\\right)} |\\mathbf{U}-\\mathbf{U}_0|", + "\\\\\\\\", + "\\ar{}+E(\\mathbf{G}(\\mathbf{U}))", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_{0}|.", + "\\end{array}", + "$$", + "Since", + "$$", + "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}E(\\mathbf{G}(\\mathbf{U}))=\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}E(\\mathbf{X})=0,", + "$$", + "\\eqref{eq:5.4.2} and Lemma~\\ref{thmtype:5.4.2} imply that", + "$$", + "\\frac{h(\\mathbf{U})-h(\\mathbf{U}_0)-\\dst\\sum_{i=1}^nf_{x_i}(\\mathbf{X}_{0}", + "d_{\\mathbf{U}_0}g_i", + "(\\mathbf{U}-\\mathbf{U}_0)}{|\\mathbf{U}-\\mathbf{U}_0|}=0.", + "$$", + "Therefore, $h$ is differentiable at $\\mathbf{U}_0$, and $d_{\\mathbf{U}_0}h$", + "is given by \\eqref{eq:5.4.4}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.3.8", + "TRENCH_REAL_ANALYSIS-thmtype:5.4.2" + ], + "ref_ids": [ + 254, + 255 + ] + } + ], + "ref_ids": [] + }, + { + "id": 164, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", + "categories": [], + "title": "Mean Value Theorem for Functions of $\\mathbf n$ Variables", + "contents": [ + "Let $f$ be continuous at $\\mathbf{X}_1=(x_{11},x_{21}, \\dots, x_{n1})$", + "and $\\mathbf{X}_2=(x_{12},x_{22}, \\dots,x_{n2})$ and differentiable on the", + "line segment $L$ from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$ Then", + "\\begin{equation} \\label{eq:5.4.21}", + "f(\\mathbf{X}_2)-f(\\mathbf{X}_1)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)(x_{i2}-x_{i1})=(d_{\\mathbf{X}_0}f)(\\mathbf{X}_2", + "-\\mathbf{X}_1)", + "\\end{equation}", + "for some $\\mathbf{X}_0$ on $L$ distinct", + "from $\\mathbf{X}_1$ and $\\mathbf{X}_2$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "An equation of $L$ is", + "$$", + "\\mathbf{X}=\\mathbf{X}(t)=t\\mathbf{X}_2+(1-t)\\mathbf{X}_1,\\quad 0\\le t\\le1.", + "$$", + "Our hypotheses imply that the function", + "$$", + "h(t)=f(\\mathbf{X}(t))", + "$$", + "is continuous on $[0,1]$ and differentiable on $(0,1)$. Since", + "$$", + "x_i(t)=tx_{i2}+(1-t)x_{i1},", + "$$", + "\\eqref{eq:5.4.20} implies that", + "$$", + "h'(t)=\\sum_{i=1}^n f_{x_i}(\\mathbf{X}(t))(x_{i2}-x_{i1}),\\quad 00$, there is a $\\delta>0$ such that", + "$B_\\delta (\\mathbf{X}_0)\\subset N$ and all $k$th-order partial", + "derivatives of $f$ satisfy the inequality", + "\\begin{equation} \\label{eq:5.4.32}", + "\\left|\\frac{\\partial^kf(\\widetilde{\\mathbf{X}})}{\\partial x_{i_k}\\partial", + "x_{i_{k-1}} \\cdots\\partial x_{i_1}}-", + "\\frac{\\partial^kf(\\mathbf{X}_0)}{\\partial x_{i_k} \\partial", + "x_{i_{k-1}}\\cdots\\partial", + "x_{i_1}}\\right|<\\epsilon,\\quad \\widetilde{\\mathbf{X}}\\in B_\\delta (\\mathbf{X}_0).", + "\\end{equation}", + " Now suppose that $\\mathbf{X}\\in B_\\delta (\\mathbf{X}_0)$. From", + "Theorem~\\ref{thmtype:5.4.8} with $k$ replaced by $k-1$,", + "\\begin{equation} \\label{eq:5.4.33}", + "f(\\mathbf{X})=T_{k-1}(\\mathbf{X})+\\frac{1}{ k!}", + "(d^{(k)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-\\mathbf{X}_0),", + "\\end{equation}", + "where $\\widetilde{\\mathbf{X}}$ is some point", + " on the line segment from $\\mathbf{X}_0$ to $\\mathbf{X}$ and is therefore", + "in $B_\\delta(\\mathbf{X}_0)$. We can rewrite \\eqref{eq:5.4.33} as", + "\\begin{equation} \\label{eq:5.4.34}", + " f(\\mathbf{X})=T_k(\\mathbf{X})+\\frac{1}{", + "k!}\\left[(d^{(k)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-\\mathbf{X}_0)-", + "(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{X}-\\mathbf{X}_0)\\right].", + "\\end{equation}", + "But \\eqref{eq:5.4.23} and", + "\\eqref{eq:5.4.32} imply that", + "\\begin{equation} \\label{eq:5.4.35}", + "\\left|(d^{(k)}_{\\widetilde{\\mathbf{X}}}f)(\\mathbf{X}-\\mathbf{X}_0)-(d^{(k)}_{{\\mathbf{X}}_0}f)(\\mathbf{X}-\\mathbf{X}_0)\\right|< n^k\\epsilon |\\mathbf{X}-\\mathbf{X}_0|^k", + "\\end{equation}", + " (Exercise~\\ref{exer:5.4.17}), which", + "implies that", + "$$", + "\\frac{|f(\\mathbf{X})-T_k(\\mathbf{X})|}", + "{ |\\mathbf{X}-\\mathbf{X}_0|^k}<\\frac{n^k\\epsilon}{ k!}, \\quad\\mathbf{X}\\in", + "B_\\delta (\\mathbf{X}_0),", + "$$", + "from \\eqref{eq:5.4.34}.", + "This implies \\eqref{eq:5.4.31}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.8" + ], + "ref_ids": [ + 165 + ] + } + ], + "ref_ids": [] + }, + { + "id": 167, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.10", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ satisfies the hypotheses of Theorem~$\\ref{thmtype:5.4.9}$", + "with $k\\ge2,$ and", + " \\begin{equation} \\label{eq:5.4.38}", + "d^{(r)}_{\\mathbf{X}_0} f\\equiv0\\quad (1\\le r\\le k-1),\\quad d^{(k)}_\\mathbf{X_0}", + "f\\not\\equiv0.", + "\\end{equation}", + "Then", + "\\begin{alist}", + "\\item % (a)", + "$\\mathbf{X}_0$ is not a local extreme point of $f$ unless $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite as a polynomial in $\\mathbf{X}-\\mathbf{X}_0.$", + "In particular$,$", + " $\\mathbf{X}_0$ is not a local extreme point of $f$ if", + "$k$ is odd$.$", + "\\item % (b)", + " $\\mathbf{X}_0$ is a local minimum point of $f$ if $d^{(k)}_{\\mathbf{X}_0}", + "f$ is positive definite$,$ or a local maximum point if $d^{(k)}_{\\mathbf{X}_0}f$ is", + "negative definite$.$", + "\\item % (c)", + " If $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite$,$ then $\\mathbf{X}_0$ may be a", + "local extreme point of $f,$ but it need not be$.$", + "\\end{alist}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.9" + ], + "proofs": [ + { + "contents": [ + "From \\eqref{eq:5.4.38} and Theorem~\\ref{thmtype:5.4.9},", + "\\begin{equation} \\label{eq:5.4.39}", + "\\lim_{ \\mathbf{X}\\to\\mathbf{X}_0}", + "\\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-\\dst\\frac{1}{k!}", + "(d^{(k)}_{\\mathbf{X}_0})(\\mathbf{X}-\\mathbf{X}_0)}{ |\\mathbf{X}-\\mathbf{X}_0|^k}=0.", + "\\end{equation}", + "If $\\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U}$, where $\\mathbf{U}$ is a constant", + "vector, then", + "$$", + "(d^{(k)}_{\\mathbf{X}_0} f) (\\mathbf{X}-\\mathbf{X}_0)=", + "t^k(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{U}),", + "$$", + "so \\eqref{eq:5.4.39} implies that", + "$$", + "\\lim_{t\\to 0} \\frac{f(\\mathbf{X}_0+t\\mathbf{U})-", + "f(\\mathbf{X}_0)-\\dst\\frac{t^k}{k!}(d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{U})}{", + "t^k}=0,", + "$$", + "or, equivalently,", + "\\begin{equation} \\label{eq:5.4.40}", + "\\lim_{t\\to 0}\\frac{f(\\mathbf{X}_0+t\\mathbf{U})-f(\\mathbf{X}_0)}{ t^k}=\\frac{1}{ k!}", + "(d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{U})", + "\\end{equation}", + "for any constant vector $\\mathbf{U}$.", + "To prove \\part{a}, suppose that", + "$d^{(k)}_{\\mathbf{X}_0}f$ is not semidefinite. Then there are vectors $\\mathbf{U}_1$ and", + "$\\mathbf{U}_2$ such that", + "$$", + "(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{U}_1)>0\\mbox{\\quad and\\quad} (d^{(k)}_\\mathbf{X_0}f)(\\mathbf{U}_2)<0.", + "$$", + "This and \\eqref{eq:5.4.40} imply that", + "$$", + "f(\\mathbf{X}_0+t\\mathbf{U}_1)>f(\\mathbf{X}_0)\\mbox{\\quad and\\quad}", + " f(\\mathbf{X}_0+t\\mathbf{U}_2)0$ such that", + "\\begin{equation} \\label{eq:5.4.41}", + "\\frac{(d^{(k)}_{\\mathbf{X}_0} f)(\\mathbf{X}-\\mathbf{X}_0)}{ k!}\\ge\\rho", + "|\\mathbf{X}-\\mathbf{X}_0|^k", + "\\end{equation}", + "\\newpage", + "\\noindent", + "for all $\\mathbf{X}$ (Exercise~\\ref{exer:5.4.19}). From \\eqref{eq:5.4.39}, there", + "is a $\\delta>0$ such that", + "$$", + "\\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-\\dst\\frac{1}{k!} (d^{(k)}_{\\mathbf{X}_0}f)(\\mathbf{X}-\\mathbf{X}_0)}{ |\\mathbf{X}-\\mathbf{X}_0|^k}>-", + "\\frac{\\rho}{2}\\mbox{\\quad if\\quad} |\\mathbf{X}-\\mathbf{X}_0|<\\delta.", + "$$", + "Therefore,", + "$$", + "f(\\mathbf{X})-f(\\mathbf{X}_0)>\\frac{1}{ k!}", + "(d^{(k)}_{\\mathbf{X}_0})(\\mathbf{X}-\\mathbf{X}_0)-\\frac{\\rho}{2}", + "|\\mathbf{X}-\\mathbf{X}_0|^k\\mbox{\\quad if\\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", + "$$", + "This and \\eqref{eq:5.4.41} imply that", + "$$", + "f(\\mathbf{X})-f(\\mathbf{X}_0)>\\frac{\\rho}{2}", + " |\\mathbf{X}-\\mathbf{X}_0|^k\\mbox{\\quad if\\quad} |\\mathbf{X}-\\mathbf{X}_0| <\\delta,", + "$$", + "which implies that $\\mathbf{X}_0$ is a local minimum point of $f$. This proves", + "half of \\part{b}. We leave the other half to you", + "(Exercise~\\ref{exer:5.4.20}).", + "To prove \\part{c} merely requires examples; see Exercise~\\ref{exer:5.4.21}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.9" + ], + "ref_ids": [ + 166 + ] + } + ], + "ref_ids": [ + 166 + ] + }, + { + "id": 168, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.2", + "categories": [], + "title": "", + "contents": [ + " A transformation $\\mathbf{L}: \\R^n \\to \\R^m$", + "defined on all of $\\R^n$ is linear if and only if", + "\\begin{equation}\\label{eq:6.1.1}", + "\\mathbf{L}(\\mathbf{X})=\\left[\\begin{array}{c} a_{11}x_1+a_{12}x_2+", + "\\cdots+a_{1n}x_n\\\\a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\\\", + "\\vdots\\\\a_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n\\end{array}\\right],", + "\\end{equation}", + "where the $a_{ij}$'s are constants$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If can be seen by induction (Exercise~\\ref{exer:6.1.1}) that if", + "$\\mathbf{L}$ is linear, then", + "\\begin{equation}\\label{eq:6.1.2}", + "\\mathbf{L}(a_1\\mathbf{X}_1+a_2\\mathbf{X}_2+\\cdots+a_k\\mathbf{X}_k)=", + "a_1\\mathbf{L}(\\mathbf{X}_1)+a_2\\mathbf{L}(\\mathbf{X}_2)+\\cdots+a_k\\mathbf{L}(\\mathbf{X}_k)", + "\\end{equation}", + "for any vectors $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_k$ and real", + "numbers", + "$a_1$, $a_2$, \\dots, $a_k$. Any $\\mathbf{X}$ in $\\R^n$ can be", + "written as", + "\\begin{eqnarray*}", + "\\mathbf{X}\\ar=\\left[\\begin{array}{c} x_1\\\\ x_2\\\\\\vdots\\\\ x_n\\end{array}\\right]", + "=x_1\\left[\\begin{array}{c} 1\\\\ 0\\\\\\vdots\\\\ 0\\end{array}\\right]", + "+x_2\\left[\\begin{array}{c} 0\\\\ 1\\\\\\vdots\\\\ 0\\end{array}\\right]+\\cdots", + "+x_n\\left[\\begin{array}{c} 0\\\\ 0\\\\\\vdots\\\\ 1\\end{array}\\right]\\\\", + "\\ar=x_1\\mathbf{E}_1+x_2\\mathbf{E}_2+\\cdots+x_n\\mathbf{E}_n.", + "\\end{eqnarray*}", + "Applying \\eqref{eq:6.1.2} with $k=n$, $\\mathbf{X}_i=\\mathbf{E}_i$, and", + "$a_i=x_i$ yields", + "\\begin{equation}\\label{eq:6.1.3}", + "\\mathbf{L}(\\mathbf{X})=x_1\\mathbf{L}(\\mathbf{E}_1)+x_2\\mathbf{L}(\\mathbf{E}_2)", + "+\\cdots+x_n\\mathbf{L}(\\mathbf{E}_n).", + "\\end{equation}", + "Now denote", + "$$", + "\\mathbf{L}(\\mathbf{E}_j)=\\left[\\begin{array}{c} a_{1j}\\\\ a_{2j}\\\\", + "\\vdots\\\\ a_{mj}\\end{array}\\right],", + "$$", + "so \\eqref{eq:6.1.3} becomes", + "$$", + "\\mathbf{L}(\\mathbf{X})=x_1\\left[\\begin{array}{c} a_{11}\\\\ a_{21}\\\\\\vdots\\\\ a_{m1}", + "\\end{array}\\right]", + "+x_2\\left[\\begin{array}{c} a_{12}\\\\ a_{22}\\\\\\vdots\\\\ a_{m2}\\end{array}", + "\\right]+\\cdots", + "+x_n\\left[\\begin{array}{c} a_{1n}\\\\ a_{2n}\\\\\\vdots\\\\ a_{mn}\\end{array}", + "\\right],", + "$$", + "which is equivalent to \\eqref{eq:6.1.1}. This proves that if $\\mathbf{L}$ is", + "linear, then $\\mathbf{L}$ has the form \\eqref{eq:6.1.1}. We leave the proof of the", + "converse to you (Exercise~\\ref{exer:6.1.2})." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 169, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.4", + "categories": [], + "title": "", + "contents": [ + " If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are", + "$m\\times n$ matrices$,$ then", + "$$", + "(\\mathbf{A}+\\mathbf{B})+\\mathbf{C}=\\mathbf{A}+(\\mathbf{B}", + "+\\mathbf{C}).", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 170, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.5", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{A}$ and $\\mathbf{B}$ are $m\\times n$", + "matrices and $r$ and $s$ are real numbers$,$ then \\part{a}", + "$r(s\\mathbf{A})", + "=(rs)\\mathbf{A};$ \\part{b} $(r+s)\\mathbf{A}=r\\mathbf{A}+s\\mathbf{A};$", + "\\part{c} $r(\\mathbf{A}+\\mathbf{B})=r\\mathbf{A}+r\\mathbf{B}.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 171, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.6", + "categories": [], + "title": "", + "contents": [ + " If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are", + "$m\\times p,$ $p\\times q,$ and $q\\times n$ matrices$,$ respectively$,$", + "then", + "$(\\mathbf{AB})\\mathbf{C}=\\mathbf{A}(\\mathbf{BC}).$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 172, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.7", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + "If we regard the vector", + "$$", + "\\mathbf{X}=\\left[\\begin{array}{c} x_1\\\\ x_2\\\\\\vdots\\\\", + "x_n\\end{array}\\right]", + "$$", + "as an $n\\times 1$ matrix$,$ then the linear transformation", + "$\\eqref{eq:6.1.1}$ can be written as", + "$$", + "\\mathbf{L}(\\mathbf{X})=\\mathbf{AX}.", + "$$", + "\\newpage", + "\\noindent", + "\\item % (b)", + "If $\\mathbf{L}_1$ and $\\mathbf{L}_2$ are linear transformations from", + "$\\R^n$ to $\\R^m$ with matrices $\\mathbf{A}_1$ and $\\mathbf{A}_{2}$", + "respectively$,$ then $c_1\\mathbf{L}_1+c_2\\mathbf{L}_2$ is the linear", + "transformation", + "from $\\R^n$ to $\\R^m$ with matrix $c_1\\mathbf{A}_1+c_2\\mathbf{A}_{2}.$", + "\\item % (c)", + "If $\\mathbf{L}_1: \\R^n\\to \\R^p$ and $\\mathbf{L}_2: \\R^p\\to", + "\\R^m$ are linear transformations with matrices $\\mathbf{A}_1$ and", + "$\\mathbf{A}_2,$ respectively$,$ then the composite function", + "$\\mathbf{L}_3=\\mathbf{L}_2\\circ\\mathbf{L}_1,$ defined by", + "$$", + "\\mathbf{L}_3(\\mathbf{X})=\\mathbf{L}_2(\\mathbf{L}_1(\\mathbf{X})),", + "$$", + "is the linear transformation from $\\R^n$ to $\\R^m$ with", + "matrix $\\mathbf{A}_2\\mathbf{A}_1.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 173, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.9", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{A}$ and $\\mathbf{B}$ are $n\\times n$ matrices$,$ then", + "$$", + "\\det(\\mathbf{A}\\mathbf{B})=\\det(\\mathbf{A})\\det(\\mathbf{B}).", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 174, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.11", + "categories": [], + "title": "", + "contents": [ + "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$", + "\\begin{alist}", + "\\item % (a)", + "The sum of the products of the entries of a row of $\\mathbf{A}$", + "and their cofactors equals $\\det(\\mathbf{A}),$ while the", + " sum of the products of the entries of a row of $\\mathbf{A}$", + "and the cofactors of the entries of a different row equals zero$;$", + "that is$,$", + "\\begin{equation} \\label{eq:6.1.8}", + "\\sum^n_{k=1} a_{ik}c_{jk}=\\left\\{\\casespace\\begin{array}{ll}\\det(\\mathbf{A}),&i=j,\\\\", + " 0,&i\\ne j.\\end{array}\\right.", + "\\end{equation}", + "\\item % (b)", + "The sum of the products of the entries of a column of $\\mathbf{A}$", + "and their cofactors equals $\\det(\\mathbf{A}),$ while the", + " sum of the products of the entries of a column of $\\mathbf{A}$", + "and the cofactors of the entries of a different column equals zero$;$", + "that is$,$", + "\\begin{equation} \\label{eq:6.1.9}", + "\\sum^n_{k=1} c_{ki}a_{kj}=\\left\\{\\casespace\\begin{array}{ll}", + "\\det(\\mathbf{A}),", + "&i=j,\\\\", + " 0,&i\\ne j.\\end{array}\\right.", + "\\end{equation}", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 175, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.12", + "categories": [], + "title": "", + "contents": [ + "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$", + "If $\\det(\\mathbf{A})=0,$ then $\\mathbf{A}$ is singular$.$ If", + "$\\det(\\mathbf{A})\\ne0,$ then $\\mathbf{A}$ is nonsingular$,$ and $\\mathbf{A}$", + "has the unique inverse", + "\\begin{equation} \\label{eq:6.1.10}", + "\\mathbf{A}^{-1}=\\frac{1}{\\det(\\mathbf{A})}\\adj(\\mathbf{A}).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $\\det(\\mathbf{A})=0$, then $\\det(\\mathbf{A}\\mathbf{B})=0$ for any $n\\times", + "n$ matrix, by Theorem~\\ref{thmtype:6.1.9}. Therefore, since", + "$\\det(\\mathbf{I})=1$,", + " there is no matrix $n\\times n$ matrix $\\mathbf{B}$ such that", + "$\\mathbf{A}\\mathbf{B}=\\mathbf{I}$; that is, $\\mathbf{A}$ is singular if", + " $\\det(\\mathbf{A})=0$.", + " Now suppose that $\\det(\\mathbf{A})\\ne0$. Since \\eqref{eq:6.1.8} implies", + "that", + "$$", + " \\mathbf{A}\\adj(\\mathbf{A})=\\det(\\mathbf{A})\\mathbf{I}", + "$$", + "and \\eqref{eq:6.1.9} implies that", + "$$", + " \\adj(\\mathbf{A})\\mathbf{A}=\\det(\\mathbf{A})\\mathbf{I},", + "$$", + "dividing both sides of these two equations by $\\det(\\mathbf{A})$", + "shows that", + " if $\\mathbf{A}^{-1}$ is as defined in \\eqref{eq:6.1.10},", + "then $\\mathbf{A}\\mathbf{A}^{-1}=\\mathbf{A}^{-1}\\mathbf{A}=\\mathbf{I}$. Therefore,", + "$\\mathbf{A}^{-1}$ is an inverse of $\\mathbf{A}$. To see that it is the only", + "inverse, suppose that $\\mathbf{B}$ is an $n\\times n$ matrix such that", + "$\\mathbf{A}\\mathbf{B}=\\mathbf{I}$. Then", + " $\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{B})=\\mathbf{A}^{-1}$,", + " so $(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{B}=\\mathbf{A}^{-1}$. Since", + "$\\mathbf{A}\\mathbf{A}^{-1}=\\mathbf{I}$ and $\\mathbf{I}\\mathbf{B}=\\mathbf{B}$, it follows", + "that $\\mathbf{B}=\\mathbf{A}^{-1}$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" + ], + "ref_ids": [ + 173 + ] + } + ], + "ref_ids": [] + }, + { + "id": 176, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.13", + "categories": [], + "title": "", + "contents": [ + "The system $\\eqref{eq:6.1.11}$ has a solution $\\mathbf{X}$ for any given", + "$\\mathbf{Y}$ if and only if $\\mathbf{A}$ is nonsingular$.$ In this case$,$", + "the", + "solution is unique and is given by $\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $\\mathbf{A}$ is nonsingular, and let", + "$\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$. Then", + "$$", + "\\mathbf{A}\\mathbf{X}=\\mathbf{A}(\\mathbf{A}^{-1}\\mathbf{Y})=", + "(\\mathbf{A}\\mathbf{A}^{-1})\\mathbf{Y}", + "=\\mathbf{I}\\mathbf{Y}=\\mathbf{Y};", + "$$", + "that is, $\\mathbf{X}$ is a solution of \\eqref{eq:6.1.11}.", + "To see that $\\mathbf{X}$ is the only solution of \\eqref{eq:6.1.11},", + "suppose that $\\mathbf{A}\\mathbf{X}_1=\\mathbf{Y}$.", + " Then $\\mathbf{A}\\mathbf{X}_1=\\mathbf{A}", + "\\mathbf{X}$, so", + "\\begin{eqnarray*}", + "\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{X})\\ar=", + "\\mathbf{A}^{-1}(\\mathbf{A}\\mathbf{X}_1)\\\\", + "\\arraytext{and}\\\\", + "(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{X}\\ar=", + "(\\mathbf{A}^{-1}\\mathbf{A})\\mathbf{X}_1,", + "\\end{eqnarray*}", + "which is equivalent to $\\mathbf{I}\\mathbf{X}=\\mathbf{I}\\mathbf{X}_1$, or", + "$\\mathbf{X}=\\mathbf{X}_1$.", + "Conversely, suppose that \\eqref{eq:6.1.11} has a solution for every", + "$\\mathbf{Y}$, and let", + " $\\mathbf{X}_i$", + "satisfy $\\mathbf{A}\\mathbf{X}_i=\\mathbf{E}_i$, $1\\le i\\le n$. Let", + "$$", + "\\mathbf{B}=", + "[\\mathbf{X}_1\\,\\mathbf{X}_2\\,\\cdots\\,\\mathbf{X}_n];", + "$$", + "that is, $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_n$ are the columns", + "of $\\mathbf{B}$. Then", + "$$", + "\\mathbf{A}\\mathbf{B}=", + "[\\mathbf{A}\\mathbf{X}_1\\,\\mathbf{A}\\mathbf{X}_2\\,\\cdots\\,\\mathbf{A}\\mathbf{X}_n]=", + "[\\mathbf{E}_1\\,\\mathbf{E}_2\\,\\cdots\\,\\mathbf{E}_n]", + "=\\mathbf{I}.", + "$$", + "To show that $\\mathbf{B}=\\mathbf{A}^{-1}$, we must still show", + "that $\\mathbf{B}\\mathbf{A}=\\mathbf{I}$. We first note that,", + "since $\\mathbf{A}\\mathbf{B}", + "=\\mathbf{I}$ and $\\det(\\mathbf{B}\\mathbf{A})=\\det(\\mathbf{A}\\mathbf{B})=1$", + "(Theorem~\\ref{thmtype:6.1.9}), $\\mathbf{B}\\mathbf{A}$ is nonsingular", + "(Theorem~\\ref{thmtype:6.1.12}). Now note that", + "$$", + "(\\mathbf{B}\\mathbf{A})(\\mathbf{B}\\mathbf{A})=", + "\\mathbf{B}(\\mathbf{A}\\mathbf{B})\\mathbf{A})=\\mathbf{B}\\mathbf{I}\\mathbf{A};", + "$$", + "that is,", + "$$", + "(\\mathbf{B}\\mathbf{A})(\\mathbf{B}\\mathbf{A})=(\\mathbf{B}\\mathbf{A}).", + "$$", + "Multiplying both sides of this equation on the left by", + "$\\mathbf{B}\\mathbf{A})^{-1}$ yields $\\mathbf{B}\\mathbf{A}=\\mathbf{I}$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.1.9", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.12" + ], + "ref_ids": [ + 173, + 175 + ] + } + ], + "ref_ids": [] + }, + { + "id": 177, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.14", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{A}=[a_{ij}]$ is nonsingular$,$ then the solution of", + " the system", + "\\begin{eqnarray*}", + "a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n\\ar=y_1\\\\", + "a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\ar=y_2\\\\", + "&\\vdots& \\\\", + "a_{n1}x_1+a_{n2}x_2+\\cdots+a_{nn}x_n\\ar=y_n", + "\\end{eqnarray*}", + "$($or$,$ in matrix form$,$ $\\mathbf{AX}=\\mathbf{Y}$$)$ is given", + "by", + "$$", + "x_i=\\frac{D_i}{\\det(\\mathbf{A})},\\quad 1\\le i\\le n,", + "$$", + "where $D_i$ is the determinant of the matrix obtained by replacing the", + "$i$th column of $\\mathbf{A}$ with $\\mathbf{Y};$ thus$,$", + "$$", + "D_1=\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\", + "y_2&a_{22}&\\dots&a_{2n}\\\\", + "\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,\\quad", + "D_2=\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\", + "a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\", + "\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,\\quad\\cdots,", + "$$", + "$$", + "D_n=\\left|\\begin{array}{cccc} a_{11}&\\cdots&a_{1,n-1}&y_1\\\\", + "a_{21}&\\cdots&a_{2,n-1}&y_2\\\\", + "\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "a_{n1}&\\cdots&a_{n,n-1}&y_n\\end{array}\\right|.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Theorems~\\ref{thmtype:6.1.12} and \\ref{thmtype:6.1.13}, the solution of", + "$\\mathbf{A}\\mathbf{X}=\\mathbf{Y}$ is", + "\\begin{eqnarray*}", + "\\left[\\begin{array}{c}", + "x_1\\\\x_2\\\\\\vdots\\\\x_n", + "\\end{array}\\right]", + "=\\mathbf{A}^{-1}\\mathbf{Y}", + "\\ar=\\frac{1}{\\det(\\mathbf{A})}", + "\\left[\\begin{array}{cccc}", + "c_{11}&c_{21}&\\cdots&c_{n1}\\\\", + "c_{12}&c_{22}&\\cdots&c_{n2}\\\\", + "\\cdots&\\cdots&\\ddots&\\cdots\\\\", + "c_{1n}&c_{2n}&\\cdots&c_{nn}", + "\\end{array}\\right]", + "\\left[\\begin{array}{c}", + "y_1\\\\y_2\\\\\\vdots\\\\y_n", + "\\end{array}\\right]\\\\", + "\\ar=", + "\\left[\\begin{array}{c}", + "c_{11}y_1+c_{21}y_2+\\cdots+c_{n1}y_n\\\\", + "c_{12}y_1+c_{22}y_2+\\cdots+c_{n2}y_n\\\\", + "\\vdots\\\\", + "c_{1n}y_1+c_{2n}y_2+\\cdots+c_{nn}y_n", + "\\end{array}\\right].", + "\\end{eqnarray*}", + "But", + "$$", + "c_{11}y_1+c_{21}y_2+\\cdots+c_{n1}y_n=", + "\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\", + "y_2&a_{22}&\\dots&a_{2n}\\\\", + "\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,", + "$$", + "\\newpage", + "\\noindent", + "as can be seen by expanding the determinant on the right", + "in cofactors of its first column. Similarly,", + "$$", + "c_{12}y_1+c_{22}y_2+\\cdots+c_{n2}y_n=", + "\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\", + "a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\", + "\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,", + "$$", + "as can be seen by expanding the determinant on the right", + "in cofactors of its second column. Continuing in this way completes", + "the proof." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.1.12", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.13" + ], + "ref_ids": [ + 175, + 176 + ] + } + ], + "ref_ids": [] + }, + { + "id": 178, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.15", + "categories": [], + "title": "", + "contents": [ + "The homogeneous system $\\eqref{eq:6.1.12}$ of $n$ equations in $n$", + "unknowns has a nontrivial solution if and only if $\\det(\\mathbf{A})=0.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 179, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.16", + "categories": [], + "title": "", + "contents": [ + "If $A_1,$ $A_2,$ \\dots$,$ $A_k$ are nonsingular $n\\times n$", + "matrices$,$ then so is $A_1A_2\\cdots A_k,$ and", + "$$", + "(A_1A_2\\cdots A_k)^{-1}=A_k^{-1}A_{k-1}^{-1}\\cdots A_1^{-1}.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 180, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.1", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{X}_0$ is in$,$ and a limit point of$,$ the domain", + "of", + "$\\mathbf{F}: \\R^n\\to\\R^m.$ Then $\\mathbf{F}$ is continuous at", + "$\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there is a $\\delta>0$", + "such that", + "\\begin{equation}\\label{eq:6.2.1}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|<\\epsilon", + "\\mbox{\\quad if \\quad} |\\mathbf{X}-\\mathbf{X}_0|<\\delta", + "\\mbox{\\quad and \\quad} \\mathbf{X}\\in D_\\mathbf{F}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 181, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.2", + "categories": [], + "title": "", + "contents": [ + "A transformation", + "$\\mathbf{F}=(f_1,f_2, \\dots,f_m)$ defined in a neighborhood of", + "$\\mathbf{X}_0\\in\\R^n$", + " is differentiable at $\\mathbf{X}_0$ if and only if", + "there is a constant $m\\times n$ matrix $\\mathbf{A}$ such that", + "\\begin{equation}\\label{eq:6.2.2}", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", + "\\frac{", + "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{A} (\\mathbf{X}-\\mathbf{X}_0)}", + "{|\\mathbf{X}-\\mathbf{X}_0|}=\\mathbf{0}.", + " \\end{equation}", + "If $\\eqref{eq:6.2.2}$ holds$,$ then $\\mathbf{A}$ is given uniquely by", + "\\begin{equation}\\label{eq:6.2.3}", + "\\mathbf{A}=\\left[\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}\\right]=", + "\\left[\\begin{array}{cccc}\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial", + "x_1}}&", + "\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_n}}\\\\", + "[3\\jot]", + "\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_1}}&", + "\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_2}}&", + "\\cdots&\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_n}}\\\\", + "\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_1}}&", + "\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x _2}}&", + "\\cdots&\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_n}}", + "\\end{array}\\right].", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $\\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0})$.", + " If $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0$, then so are", + "$f_1$, $f_2$, \\dots, $f_m$ (Definition~\\ref{thmtype:5.4.1}).", + "Hence,", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{\\dst{f_i(\\mathbf{X})-f_i(\\mathbf{X}_0) -", + "\\sum_{j=1}^n \\frac{\\partial", + "f_i(\\mathbf{X}_0)}{\\partial x_j} (x_j-x_{j0})}}", + "{ |\\mathbf{X}-\\mathbf{X}_{0}|}=0,", + "\\quad 1\\le i\\le m,", + "$$", + "which implies \\eqref{eq:6.2.2} with $\\mathbf{A}$ as in", + "\\eqref{eq:6.2.3}.", + "Now suppose that \\eqref{eq:6.2.2} holds", + "with $\\mathbf{A}=[a_{ij}]$. Since", + "each component of the vector in \\eqref{eq:6.2.2}", + " approaches zero as $\\mathbf{X}$", + " approaches $\\mathbf{X}_0$, it follows that", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", + " \\frac{\\dst{f_i(\\mathbf{X})-f_i(\\mathbf{X}_0)", + "-\\dst{\\sum_{j=1}^n} a_{ij}", + "(x_j-x_{j0})}}{ |\\mathbf{X}-\\mathbf{X}_0|}", + "=0,\\quad 1\\le i\\le m,", + "$$", + "so each $f_i$ is differentiable at $\\mathbf{X}_0$, and therefore so", + "is $\\mathbf{F}$ (Definition~\\ref{thmtype:5.4.1}).", + "By Theorem~\\ref{thmtype:5.3.6},", + "$$", + "a_{ij}=\\frac{\\partial f_i (\\mathbf{X}_0)}{\\partial x_j},\\quad 1\\le i\\le m,", + "\\quad 1\\le j\\le n,", + "$$", + "which implies \\eqref{eq:6.2.3}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", + "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", + "TRENCH_REAL_ANALYSIS-thmtype:5.3.6" + ], + "ref_ids": [ + 351, + 351, + 158 + ] + } + ], + "ref_ids": [] + }, + { + "id": 182, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.3", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at", + "$\\mathbf{X}_0,$ then $\\mathbf{F}$ is continuous at~$\\mathbf{X}_0.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 183, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.4", + "categories": [], + "title": "", + "contents": [ + "Let $\\mathbf{F}=(f_1,f_2, \\dots,f_m):\\R^n\\to\\R^m,$ and", + "suppose that the partial derivatives", + "\\begin{equation}\\label{eq:6.2.7}", + "\\frac{\\partial f_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le", + "n,", + "\\end{equation}", + "exist on a neighborhood of $\\mathbf{X}_0$ and", + "are continuous at $\\mathbf{X}_0.$ Then $\\mathbf{F}$ is differentiable at", + "$\\mathbf{X}_0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Consider the auxiliary function", + "\\begin{equation} \\label{eq:6.2.9}", + "\\mathbf{G}(\\mathbf{X})=\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}_0)\\mathbf{X}.", + "\\end{equation}", + "The components of $\\mathbf{G}$ are", + "$$", + "g_i(\\mathbf{X})=f_i(\\mathbf{X})-\\sum_{j=1}^n", + "\\frac{\\partial f_i(\\mathbf{X}_{0})", + "\\partial x_j} x_j,", + "$$", + "so", + "$$", + "\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}=", + "\\frac{\\partial f_i(\\mathbf{X})}", + "{\\partial x_j}-\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}.", + "$$", + "\\newpage", + "\\noindent", + "Thus, $\\partial g_i/\\partial x_j$ is continuous on $N$ and zero at", + "$\\mathbf{X}_0$. Therefore, there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:6.2.10}", + "\\left|\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}\\right|<\\frac{\\epsilon}{", + "\\sqrt{mn}}\\mbox{\\quad for \\quad}1\\le i\\le m,\\quad 1\\le j\\le n,", + "\\mbox{\\quad if \\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", + "\\end{equation}", + "Now suppose that $\\mathbf{X}$, $\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0)$. By", + "Theorem~\\ref{thmtype:5.4.5},", + "\\begin{equation}\\label{eq:6.2.11}", + "g_i(\\mathbf{X})-g_i(\\mathbf{Y})=\\sum_{j=1}^n", + "\\frac{\\partial g_i(\\mathbf{X}_i)}{\\partial x_j}(x_j-y_j),", + "\\end{equation}", + "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$,", + "so $\\mathbf{X}_i\\in B_\\delta(\\mathbf{X}_0)$. From \\eqref{eq:6.2.10},", + "\\eqref{eq:6.2.11}, and Schwarz's inequality,", + "$$", + "(g_i(\\mathbf{X})-g_i(\\mathbf{Y}))^2\\le\\left(\\sum_{j=1}^n\\left[\\frac{\\partial", + "g_i", + "(\\mathbf{X}_i)}{\\partial x_j}\\right]^2\\right)", + "|\\mathbf{X}-\\mathbf{Y}|^2", + "<\\frac{\\epsilon^2}{ m} |\\mathbf{X}-\\mathbf{Y}|^2.", + "$$", + "Summing this from $i=1$ to $i=m$ and taking square roots yields", + "\\begin{equation}\\label{eq:6.2.12}", + "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|<\\epsilon", + "|\\mathbf{X}-\\mathbf{Y}|", + "\\mbox{\\quad if\\quad}\\mathbf{X}, \\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", + "\\end{equation}", + "To complete the proof, we note that", + "\\begin{equation}\\label{eq:6.2.13}", + "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})=", + "\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})+\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y}),", + "\\end{equation}", + " so \\eqref{eq:6.2.12} and the triangle inequality imply \\eqref{eq:6.2.8}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.5" + ], + "ref_ids": [ + 164 + ] + } + ], + "ref_ids": [] + }, + { + "id": 184, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.8", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at", + "$\\mathbf{X}_0,$ $\\mathbf{G}:\\R^k\\to\\R^n$ is differentiable at", + "$\\mathbf{U}_0,$ and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then the composite", + "function $\\mathbf{H}=\\mathbf{F}\\circ\\mathbf{G}:\\R^k\\to\\R^m,$", + "defined by", + "$$", + "\\mathbf{H}(\\mathbf{U})=\\mathbf{F}(\\mathbf{G}(\\mathbf{U})),", + "$$", + "is differentiable at $\\mathbf{U}_0.$ Moreover$,$", + "\\begin{equation}\\label{eq:6.2.22}", + "\\mathbf{H}'(\\mathbf{U}_0)=\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))", + "\\mathbf{G}'(\\mathbf{U}_0)", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:6.2.23}", + "d_{\\mathbf{U}_0}\\mathbf{H}=d_{\\mathbf{X}_0}\\mathbf{F}\\circ d_{\\mathbf{U}_0}\\mathbf{G},", + "\\end{equation}", + "where $\\circ$ denotes composition$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The components of $\\mathbf{H}$ are $h_1$, $h_2$, \\dots, $h_m$, where", + "$$", + "h_i(\\mathbf{U})=f_i(\\mathbf{G}(\\mathbf{U})).", + "$$", + "Applying Theorem~\\ref{thmtype:5.4.3} to $h_i$ yields", + "\\begin{equation}\\label{eq:6.2.24}", + "d_{\\mathbf{U}_0}h_i=\\sum_{j=1}^n \\frac{\\partial f_i(\\mathbf{X}_{0})}", + "{\\partial x_j} d_{\\mathbf{U}_0}g_j,\\quad 1\\le i\\le m.", + "\\end{equation}", + "\\newpage", + "\\enlargethispage{\\baselineskip}", + "\\noindent Since", + "$$", + "d_{\\mathbf{U}_0}\\mathbf{H}=\\left[\\begin{array}{c}", + "d_{\\mathbf{U}_0}h_1\\\\ d_{\\mathbf{U}_0}h_2\\\\", + "\\vdots\\\\", + "d_{\\mathbf{U}_0} h_m\\end{array}\\right]\\mbox{", + "\\quad and\\quad} d_{\\mathbf{U}_0}\\mathbf{G}=", + "\\left[\\begin{array}{c} d_{\\mathbf{U}_0}g_1\\\\ d_{\\mathbf{U}_0}g_2\\\\", + "\\vdots\\\\ d_{\\mathbf{U}_0}g_n", + "\\end{array}\\right],", + "$$", + "\\vskip5pt", + "\\noindent the $m$ equations in \\eqref{eq:6.2.24} can be", + "written in matrix form as", + "\\begin{equation}\\label{eq:6.2.25}", + "d_{\\mathbf{U}_0}\\mathbf{H}=\\mathbf{F}'(\\mathbf{X}_0)d_{\\mathbf{U}_0}\\mathbf{G}=", + "\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0)) d_{\\mathbf{U}_0}\\mathbf{G}.", + "\\end{equation}", + "But", + "$$", + "d_{\\mathbf{U}_0}\\mathbf{G}=\\mathbf{G}'(\\mathbf{U}_0)\\,d\\mathbf{U},", + "$$", + "where", + "$$", + "d\\mathbf{U}=\\left[\\begin{array}{c} du_1\\\\ du_2\\\\\\vdots\\\\", + "du_k\\end{array}\\right],", + "$$", + "so \\eqref{eq:6.2.25} can be rewritten as", + "$$", + "d_{\\mathbf{U}_0}\\mathbf{H}=", + "\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))", + "\\mathbf{G}'(\\mathbf{U}_0)\\,d\\mathbf{U}.", + "$$", + "On the other hand,", + "$$", + "d_{\\mathbf{U}_0}\\mathbf{H}=\\mathbf{H}'(\\mathbf{U}_0)\\,d\\mathbf{U}.", + "$$", + "Comparing the last two equations yields \\eqref{eq:6.2.22}.", + "Since $\\mathbf{G}'(\\mathbf{U}_0)$ is the matrix of $d_{\\mathbf{U}_0}\\mathbf{G}$", + "and $\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0))=\\mathbf{F}'(\\mathbf{X}_0)$ is the matrix", + "of $d_{\\mathbf{X}_0}\\mathbf{F}$, Theorem~\\ref{thmtype:6.1.7}\\part{c}", + "and", + "\\eqref{eq:6.2.22} imply~\\eqref{eq:6.2.23}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.3", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.7" + ], + "ref_ids": [ + 163, + 172 + ] + } + ], + "ref_ids": [] + }, + { + "id": 185, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.1", + "categories": [], + "title": "", + "contents": [ + "The linear transformation", + "$$", + "\\mathbf{U}=\\mathbf{L}(\\mathbf{X})=\\mathbf{A}\\mathbf{X}\\quad (\\R^n\\to", + "\\R^n)", + "$$", + "is invertible if and only if $\\mathbf{A}$ is nonsingular$,$ in which case", + "$R(\\mathbf{L})= \\R^n$ and", + "$$", + "\\mathbf{L}^{-1}(\\mathbf{U})=\\mathbf{A}^{-1}\\mathbf{U}.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 186, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.3", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{F}: \\R^n\\to \\R^n$ is regular on an open", + "set $S,$ and let $\\mathbf{G}=\\mathbf{F}^{-1}_S.$ Then $\\mathbf{F}(S)$ is", + "open$,$", + "$\\mathbf{G}$ is continuously differentiable on $\\mathbf{F}(S),$ and", + "$$", + "\\mathbf{G}'(\\mathbf{U})=(\\mathbf{F}'(\\mathbf{X}))^{-1},", + "\\mbox{\\quad where\\quad}\\mathbf{U}=\\mathbf{F}(\\mathbf{X}).", + "$$", + "Moreover$,$ since $\\mathbf{G}$ is one-to-one on $\\mathbf{F}(S),$", + " $\\mathbf{G}$ is regular on $\\mathbf{F}(S).$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We first show that if $\\mathbf{X}_{0} \\in S$,", + " then a neighborhood of $\\mathbf{F}(\\mathbf{X}_0)$ is in", + "$\\mathbf{F}(S)$.", + "This implies that $\\mathbf{F}(S)$ is open.", + "Since $S$ is open, there is a $\\rho>0$ such that", + " $\\overline{B_\\rho(\\mathbf{X}_0)}\\subset S$. Let $B$", + "be the boundary of $B_\\rho(\\mathbf{X}_0)$; thus,", + "\\begin{equation} \\label{eq:6.3.20}", + "B=\\set\\mathbf{X}{|\\mathbf{X}-\\mathbf{X}_0|=\\rho}.", + "\\end{equation}", + "The function", + "$$", + "\\sigma(\\mathbf{X})=|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|", + "$$", + "is continuous on $S$ and therefore on $B$, which is compact. Hence,", + "by Theorem~\\ref{thmtype:5.2.12}, there is a point $\\mathbf{X}_1$", + "in $B$ where $\\sigma(\\mathbf{X})$ attains its minimum value, say $m$, on", + "$B$. Moreover, $m>0$, since $\\mathbf{X}_1\\ne\\mathbf{X}_0$ and $\\mathbf{F}$ is", + "one-to-one on $S$. Therefore,", + "\\begin{equation} \\label{eq:6.3.21}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|\\ge m>0\\mbox{\\quad if\\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|=\\rho.", + "\\end{equation}", + "The set", + "$$", + "\\set{\\mathbf{U}}{|\\mathbf{U}-\\mathbf{F}(\\mathbf{X}_0)|0$", + "and an open neighborhood $N$ of $\\mathbf{X}_0$ such that $N\\subset S$ and", + "\\begin{equation} \\label{eq:6.3.24}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|\\ge\\lambda |\\mathbf{X}-\\mathbf{X}_0|", + "\\mbox{\\quad if\\quad}\\mathbf{X}\\in N.", + "\\end{equation}", + "(Exercise~\\ref{exer:6.2.18} also implies this.) Since $\\mathbf{F}$", + "satisfies the hypotheses of the present theorem on $N$, the first part", + "of this proof shows that $\\mathbf{F}(N)$ is an open set containing", + "$\\mathbf{U}_0=\\mathbf{F} (\\mathbf{X}_0)$. Therefore, there is a", + "$\\delta>0$ such that", + "$\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ is in $N$ if $\\mathbf{U}\\in", + "B_\\delta(\\mathbf{U}_{0})$.", + " Setting $\\mathbf{X}=\\mathbf{G}(\\mathbf{U})$ and $\\mathbf{X}_0 =", + "\\mathbf{G}(\\mathbf{U}_0)$ in \\eqref{eq:6.3.24} yields", + "$$", + "|\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))-\\mathbf{F}(\\mathbf{G}(\\mathbf{U}_0))", + "|\\ge\\lambda", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|\\mbox{\\quad if \\quad}", + "\\mathbf{U}\\in B_\\delta (\\mathbf{U}_0).", + "$$", + "Since $\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))=\\mathbf{U}$, this can be rewritten as", + "\\begin{equation} \\label{eq:6.3.25}", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|\\le\\frac{1}{\\lambda} |\\mathbf{U}-", + "\\mathbf{U}_0|\\mbox{\\quad if\\quad}\\mathbf{U}\\in B_\\delta(\\mathbf{U}_0),", + "\\end{equation}", + "which means that $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$.", + "Since $\\mathbf{U}_0$ is an arbitrary point in $\\mathbf{F}(S)$, it follows", + "that $\\mathbf{G}$ is continous on $\\mathbf{F}(S)$.", + "We will now show that $\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$.", + "Since", + "$$", + "\\mathbf{G}(\\mathbf{F}(\\mathbf{X}))=\\mathbf{X},\\quad\\mathbf{X}\\in S,", + "$$", + "the chain rule (Theorem~\\ref{thmtype:6.2.8}) implies that", + "{\\it if\\/} $\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$, then", + "$$", + "\\mathbf{G}'(\\mathbf{U}_0)\\mathbf{F}'(\\mathbf{X}_0)=\\mathbf{I}", + "$$", + "\\newpage", + "\\noindent", + "(Example~\\ref{example:6.2.3}).", + " Therefore, if", + "$\\mathbf{G}$ is differentiable at $\\mathbf{U}_0$, the differential matrix", + "of $\\mathbf{G}$", + "must be", + "$$", + "\\mathbf{G}'(\\mathbf{U}_0)=[\\mathbf{F}'(\\mathbf{X}_0)]^{-1},", + "$$", + "so to show that $\\mathbf{G}$ is differentiable at", + "$\\mathbf{U}_0$, we must show that if", + "\\begin{equation} \\label{eq:6.3.26}", + "\\mathbf{H}(\\mathbf{U})=", + "\\frac{\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)-", + "[\\mathbf{F}'(\\mathbf{X}", + "_0)]^{-1} (\\mathbf{U}-\\mathbf{U}_0)}{ |\\mathbf{U}-\\mathbf{U}_0|}\\quad", + "(\\mathbf{U}\\ne\\mathbf{U}_0),", + "\\end{equation}", + "then", + "\\begin{equation} \\label{eq:6.3.27}", + "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{H}(\\mathbf{U})=\\mathbf{0}.", + "\\end{equation}", + "Since $\\mathbf{F}$ is one-to-one on $S$ and $\\mathbf{F}", + "(\\mathbf{G}(\\mathbf{U}))", + "=\\mathbf{U}$, it follows that if $\\mathbf{U}\\ne\\mathbf{U}_0$, then", + "$\\mathbf{G}(\\mathbf{U})\\ne\\mathbf{G}(\\mathbf{U}_0)$. Therefore, we can multiply", + "the numerator and denominator of \\eqref{eq:6.3.26}", + " by $|\\mathbf{G}(\\mathbf{U})", + "-\\mathbf{G}(\\mathbf{U}_0)|$ to obtain", + "$$", + "\\begin{array}{rcl}", + "\\mathbf{H}(\\mathbf{U})\\ar=", + "\\dst\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_{0}|}", + "{|\\mathbf{U}-\\mathbf{U}_0|}", + "\\left(\\frac{\\mathbf{G}(\\mathbf{U})-\\mathbf{G}", + "(\\mathbf{U}_0)-", + "\\left[\\mathbf{F}'(\\mathbf{X}_{0})", + "\\right]^{-1}(\\mathbf{U}-\\mathbf{U}_0)}", + "{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right)\\\\\\\\", + "\\ar=-\\dst\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}{", + "|\\mathbf{U}-\\mathbf{U}_0|}", + "\\left[\\mathbf{F}'(\\mathbf{X}_0)\\right]^{-1}", + "\\left(\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", + "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))", + "}{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right)", + "\\end{array}", + "$$", + " if $0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta$.", + "Because of \\eqref{eq:6.3.25}, this implies that", + "$$", + "|\\mathbf{H}(\\mathbf{U})|\\le\\frac{1}{\\lambda}", + "\\|[\\mathbf{F}'(\\mathbf{X}_0)]^{-1}\\|", + "\\left|\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", + "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))}{", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}\\right|", + "$$", + " if $0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta$.", + "Now let", + "$$", + "\\mathbf{H}_1(\\mathbf{U})=\\frac{\\mathbf{U}-\\mathbf{U}_0-\\mathbf{F}'(\\mathbf{X}_0)", + "(\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0))}{", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}", + "$$", + "To complete the proof of \\eqref{eq:6.3.27}, we must show that", + "\\begin{equation} \\label{eq:6.3.28}", + "\\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{H}_1(\\mathbf{U})=\\mathbf{0}.", + "\\end{equation}", + "Since $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0$, we know that if", + "$$", + "\\mathbf{H}_2(\\mathbf{X})=", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}", + "\\frac{\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{F}'(\\mathbf{X}_0)", + "(\\mathbf{X}-\\mathbf{X}_0)}{", + "|\\mathbf{X}-\\mathbf{X}_0|},", + "$$", + "then", + "\\begin{equation} \\label{eq:6.3.29}", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0}\\mathbf{H}_2(\\mathbf{X})=\\mathbf{0}.", + "\\end{equation}", + "Since $\\mathbf{F}(\\mathbf{G}(\\mathbf{U}))=\\mathbf{U}$ and $\\mathbf{X}_0=", + "\\mathbf{G}(\\mathbf{U}_0)$,", + "$$", + "\\mathbf{H}_1(\\mathbf{U})=\\mathbf{H}_2(\\mathbf{G}(\\mathbf{U})).", + "$$", + "\\newpage", + "\\noindent", + "Now suppose that $\\epsilon>0$. From \\eqref{eq:6.3.29}, there is a", + "$\\delta_1>0$ such that", + "\\begin{equation} \\label{eq:6.3.30}", + "|\\mathbf{H}_2(\\mathbf{X})|<\\epsilon\\mbox{\\quad if \\quad} 0<", + "|\\mathbf{X}-\\mathbf{X}_{0}|", + "=|\\mathbf{X}-\\mathbf{G}(\\mathbf{U}_0)|<\\delta_1.", + "\\end{equation}", + "Since $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$, there is a", + "$\\delta_2\\in(0,\\delta)$ such that", + "$$", + "|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\delta_1\\mbox{\\quad if \\quad}", + "0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta_2.", + "$$", + "This and \\eqref{eq:6.3.30} imply", + "that", + "$$", + "|\\mathbf{H}_1(\\mathbf{U})|=|\\mathbf{H}_2(\\mathbf{G}(\\mathbf{U}))|<\\epsilon", + "\\mbox{\\quad if \\quad} 0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta_2.", + "$$", + "Since this implies", + "\\eqref{eq:6.3.28}, $\\mathbf{G}$", + "is differentiable at $\\mathbf{X}_0$.", + "Since $\\mathbf{U}_0$ is an arbitrary member of $\\mathbf{F}(N)$, we", + "can now drop the zero subscript and conclude that $\\mathbf{G}$", + "is continuous and differentiable on $\\mathbf{F}(N)$, and", + "$$", + "\\mathbf{G}'(\\mathbf{U})=[\\mathbf{F}'(\\mathbf{X})]^{-1},\\quad\\mathbf{U}\\in\\mathbf{F}(N).", + "$$", + "To see that $\\mathbf{G}$ is \\emph{continuously differentiable} on", + "$\\mathbf{F}(N)$, we observe that by", + "Theorem~\\ref{thmtype:6.1.14}, each", + "entry of $\\mathbf{G}'(\\mathbf{U})$ (that is, each partial derivative", + "$\\partial g_i(\\mathbf{U})/\\partial u_j$, $1\\le i, j\\le n$) can be written", + "as the ratio, with nonzero denominator, of determinants with", + "entries of the form", + "\\begin{equation} \\label{eq:6.3.31}", + "\\frac{\\partial f_r(\\mathbf{G}(\\mathbf{U}))}{\\partial x_s}.", + "\\end{equation}", + "Since $\\partial f_r/\\partial x_s$ is continuous on $N$ and $\\mathbf{G}$", + "is continuous on $\\mathbf{F}(N)$, Theorem~\\ref{thmtype:5.2.10}", + "implies that \\eqref{eq:6.3.31} is continuous on $\\mathbf{F}(N)$. Since a", + "determinant is a continuous function of its entries, it now follows", + "that the entries of $\\mathbf{G}'(\\mathbf{U})$ are continuous on", + "$\\mathbf{F}(N)$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", + "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", + "TRENCH_REAL_ANALYSIS-thmtype:5.3.11", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.13", + "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", + "TRENCH_REAL_ANALYSIS-thmtype:6.2.8", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.14", + "TRENCH_REAL_ANALYSIS-thmtype:5.2.10" + ], + "ref_ids": [ + 152, + 152, + 162, + 176, + 257, + 184, + 177, + 150 + ] + } + ], + "ref_ids": [] + }, + { + "id": 187, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.4", + "categories": [], + "title": "The Inverse Function Theorem", + "contents": [ + "Let $\\mathbf{F}: \\R^n\\to \\R^n$ be continuously", + "differentiable on an open set $S,$ and", + "suppose that $J\\mathbf{F}(\\mathbf{X})\\ne0$ on $S.$ Then$,$ if $\\mathbf{X}_0\\in S,$", + "there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is", + "regular$.$ Moreover$,$ $\\mathbf{F}(N)$ is open and $\\mathbf{G}=", + "\\mathbf{F}^{-1}_N$ is continuously differentiable on $\\mathbf{F}(N),$", + "with", + "$$", + "\\mathbf{G}'(\\mathbf{U})=\\left[\\mathbf{F}'(\\mathbf{X})\\right]^{-1}\\quad", + "\\mbox{ $($where", + "$\\mathbf{U}=\\mathbf{F}(\\mathbf{X})$$)$},\\quad \\mathbf{U}\\in\\mathbf{F}(N).", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Lemma~\\ref{thmtype:6.2.6} implies that there is an open neighborhood", + "$N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is one-to-one. The rest of the", + "conclusions then follow from applying Theorem~\\ref{thmtype:6.3.3}", + " to $\\mathbf{F}$", + " on $N$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", + "TRENCH_REAL_ANALYSIS-thmtype:6.3.3" + ], + "ref_ids": [ + 257, + 186 + ] + } + ], + "ref_ids": [] + }, + { + "id": 188, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.4.1", + "categories": [], + "title": "The Implicit Function Theorem", + "contents": [ + "Suppose that $\\mathbf{F}:\\R^{n+m}\\to \\R^m$ is continuously", + "differentiable on an open set $S$ of $\\R^{n+m}$ containing", + "$(\\mathbf{X}_0,\\mathbf{U}_0).$ Let $\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0},$", + "and suppose that $\\mathbf{F}_\\mathbf{U}(\\mathbf{X}_0,\\mathbf{U}_0)$ is", + "nonsingular$.$ Then there is a neighborhood $M$ of", + " $(\\mathbf{X}_0,\\mathbf{U}_{0}),$", + " contained in $S,$ on which", + " $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$", + " is nonsingular", + " and a neighborhood $N$ of $\\mathbf{X}_0$ in", + "$\\R^n$ on which a unique continuously differentiable", + " transformation", + "$\\mathbf{G}:", + "\\R^n\\to", + "\\R^m$ is defined$,$ such that", + "$\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$ and", + "\\begin{equation} \\label{eq:6.4.6}", + "(\\mathbf{ X},\\mathbf{G}(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}", + "\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=0\\mbox{\\quad", + " if}\\quad\\mathbf{X}\\in N.", + "\\end{equation}", + "Moreover$,$", + "\\begin{equation} \\label{eq:6.4.7}", + "\\mathbf{G}'(\\mathbf{X})=-[\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))]^{-1}", + "\\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})),\\quad \\mathbf{X}\\in N.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Define $\\boldsymbol{\\Phi}:\\R^{n+m}\\to \\R^{n+m}$ by", + "\\begin{equation} \\label{eq:6.4.8}", + "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=\\left[\\begin{array}{c} x_1\\\\", + "x_2\\\\\\vdots\\\\ x_n\\\\ f_1(\\mathbf{X},\\mathbf{U})\\\\", + "[3\\jot]", + "f_2(\\mathbf{X},\\mathbf{U})\\\\\\vdots\\\\ f_m(\\mathbf{X},\\mathbf{U})\\end{array}", + "\\right]", + "\\end{equation}", + "or, in ``horizontal''notation by", + "\\begin{equation} \\label{eq:6.4.9}", + "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{U})).", + "\\end{equation}", + "Then $\\boldsymbol{\\Phi}$ is continuously differentiable on $S$ and, since", + "$\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0}$,", + "\\begin{equation} \\label{eq:6.4.10}", + "\\boldsymbol{\\Phi}(\\mathbf{X}_0,\\mathbf{U}_0)=(\\mathbf{X}_0,\\mathbf{0}).", + "\\end{equation}", + "The differential matrix of $\\boldsymbol{\\Phi}$ is", + "$$", + "\\boldsymbol{\\Phi}'=\\left[\\begin{array}{cccccccc}", + "1&0&\\cdots&0&0&0&\\cdots&0\\\\", + "[3\\jot]", + "0&1&\\cdots&0&0&0&\\cdots&0\\\\", + "\\vdots&\\vdots&\\ddots&\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "0&0&\\cdots&1&0&0&\\cdots&0\\\\", + "[3\\jot]", + "\\dst{\\frac{\\partial f_1}{\\partial x_1}}&", + "\\dst{\\frac{\\partial f_1}{\\partial x_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_1}{\\partial x_n}}&", + "\\dst{\\frac{\\partial f_1}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_1}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_1}{\\partial u_m}}\\\\", + "[3\\jot]", + "\\dst{\\frac{\\partial f_2}{\\partial x_1}}&", + "\\dst{\\frac{\\partial f_2}{\\partial x_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_2}{\\partial x_n}}&", + "\\dst{\\frac{\\partial f_2}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_2}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_2}{\\partial u_m}}\\\\", + "[3\\jot]", + "\\vdots&\\vdots&\\ddots&\\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "[3\\jot]", + "\\dst{\\frac{\\partial f_m}{\\partial x_1}}&", + "\\dst{\\frac{\\partial f_m}{\\partial x_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_m}{\\partial x_n}}&", + "\\dst{\\frac{\\partial f_m}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_m}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_m}{\\partial u_m}}\\end{array}\\right]=", + "\\left[\\begin{array}{cc}\\mathbf{I}&\\mathbf{0}\\\\\\mathbf{F}_\\mathbf{X}&\\mathbf{F}_\\mathbf{U}", + "\\end{array}\\right],", + "$$", + "\\newpage", + "\\noindent", + "where $\\mathbf{I}$ is the $n\\times n$ identity matrix, $\\mathbf{0}$ is the", + "$n\\times m$ matrix with all zero entries, and $\\mathbf{F}_\\mathbf{X}$ and", + "$\\mathbf{F}_\\mathbf{U}$ are as in \\eqref{eq:6.4.5}. By expanding", + "$\\det(\\boldsymbol{\\Phi}')$ and the determinants that evolve from it in terms", + "of the cofactors of their first rows, it can be shown in $n$ steps", + "that", + "\\vskip.5pc", + "$$", + "J\\boldsymbol{\\Phi}=\\det(\\boldsymbol{\\Phi}')=\\left|\\begin{array}{cccc}", + "\\dst{\\frac{\\partial f_1}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_1}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_1}{\\partial u_m}}\\\\", + "[3\\jot]", + "\\dst{\\frac{\\partial f_2}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_2}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_2}{\\partial u_m}}\\\\", + "[3\\jot]", + "\\vdots&\\vdots&\\ddots&\\vdots\\\\", + "\\dst{\\frac{\\partial f_m}{\\partial u_1}}&", + "\\dst{\\frac{\\partial f_m}{\\partial u_2}}&\\cdots&", + "\\dst{\\frac{\\partial f_m}{\\partial u_m}}\\end{array}\\right|=", + "\\det(\\mathbf{F}_\\mathbf{U}).", + "$$", + "\\vskip.5pc", + "In particular,", + "$$", + "J\\boldsymbol{\\Phi}(\\mathbf{X}_0,\\mathbf{U}_0)=\\det(\\mathbf{F}_\\mathbf{U}", + "(\\mathbf{X}_0,\\mathbf{U}_{0})\\ne0.", + "$$", + "Since $\\boldsymbol{\\Phi}$ is continuously differentiable on $S$,", + "Corollary~\\ref{thmtype:6.3.5} implies that $\\boldsymbol{\\Phi}$ is regular", + "on some open neighborhood $M$ of $(\\mathbf{X}_0,\\mathbf{U}_0)$ and that", + "$\\widehat{M}=\\boldsymbol{\\Phi}(M)$ is open.", + "Because of the form of $\\boldsymbol{\\Phi}$ (see \\eqref{eq:6.4.8} or", + "\\eqref{eq:6.4.9}),", + "we can write points of $\\widehat{M}$ as $(\\mathbf{X},\\mathbf{V})$,", + " where $\\mathbf{V}\\in \\R^m$.", + "Corollary~\\ref{thmtype:6.3.5} also", + "implies that $\\boldsymbol{\\Phi}$ has a a continuously differentiable", + "inverse $\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})$", + "defined on $\\widehat{M}$", + "with values in $M$. Since $\\boldsymbol{\\Phi}$ leaves the ``$\\mathbf{X}$", + "part\"", + "of $(\\mathbf{X},\\mathbf{U})$ fixed, a local inverse of $\\boldsymbol{\\Phi}$", + "must also have this property.", + " Therefore, $\\boldsymbol{\\Gamma}$ must", + "have the form", + "\\vskip.5pc", + "$$", + "\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})=\\left[\\begin{array}{c} x_1\\\\", + "x_2\\\\\\vdots\\\\ x_n\\\\[3\\jot]", + "h_1(\\mathbf{X},\\mathbf{V})\\\\[3\\jot] h_2(\\mathbf{X},\\mathbf{V})\\\\", + "\\vdots\\\\", + "[3\\jot]", + "h_m(\\mathbf{X},\\mathbf{V})\\end{array}\\right]", + "$$", + "\\vskip1pc", + "\\noindent or, in ``horizontal'' notation,", + "$$", + "\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{V})=(\\mathbf{X},\\mathbf{H}(\\mathbf{X},\\mathbf{V})),", + "$$", + "\\noindent where $\\mathbf{H}:\\R^{n+m}\\to \\R^m$ is continuously", + "differentiable on $\\widehat{M}$.", + "We will show that", + "$\\mathbf{G}(\\mathbf{X})=\\mathbf{H}(\\mathbf{X},\\mathbf{0})$", + "has the stated properties.", + "\\enlargethispage{.5\\baselineskip}", + "From \\eqref{eq:6.4.10}, $(\\mathbf{X}_0,\\mathbf{0})\\in\\widehat{M}$ and, since", + "$\\widehat{M}$ is open, there is a neighborhood $N$ of $\\mathbf{X}_0$ in", + "$\\R^n$ such that $(\\mathbf{X},\\mathbf{0})\\in\\widehat{M}$ if $\\mathbf{X}\\in", + "N$ (Exercise~\\ref{exer:6.4.2}).", + "Therefore, $(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))", + "=\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0})\\in M$ if $\\mathbf{X}\\in N$.", + " Since $\\boldsymbol{\\Gamma}=\\boldsymbol{\\Phi}^{-1}$,", + "$(\\mathbf{X},\\mathbf{0})", + "=\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$. Setting", + "$\\mathbf{X}=\\mathbf{X}_0$ and recalling \\eqref{eq:6.4.10}", + "shows that $\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$, since $\\boldsymbol{\\Phi}$", + "is one-to-one on $M$.", + "\\newpage", + "Henceforth we assume that $\\mathbf{X}\\in N$.", + "Now,", + "$$", + "\\begin{array}{rcll}", + "(\\mathbf{X},\\mathbf{0})\\ar=", + "\\boldsymbol{\\Phi}(\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0}))", + "&\\mbox{", + "(since", + "$\\boldsymbol{\\Phi}=\\boldsymbol{\\Gamma}^{-1})$}\\\\", + "\\ar=\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))&\\mbox{ (since", + "$\\boldsymbol{\\Gamma}(\\mathbf{X},\\mathbf{0})=(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$)}\\\\", + "\\ar=(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})))&\\mbox{ (since", + "$\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{U})=", + "(\\mathbf{X},\\mathbf{F}(\\mathbf{X},\\mathbf{U} ))$)}.", + "\\end{array}", + "$$", + "Therefore, $\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=\\mathbf{0}$; that is,", + "$\\mathbf{G}$ satisfies", + "\\eqref{eq:6.4.6}.", + "To see that $\\mathbf{G}$ is unique,", + "suppose that $\\mathbf{G}_1:\\R^n\\to \\R^m$ also satisfies", + "\\eqref{eq:6.4.6}. Then", + "$$", + "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=", + "(\\mathbf{X},\\mathbf{F}", + "(\\mathbf{X},\\mathbf{G}(\\mathbf{X})))=(\\mathbf{X},\\mathbf{0})", + "$$", + "and", + "$$", + "\\boldsymbol{\\Phi}(\\mathbf{X},\\mathbf{G}_1(\\mathbf{X}))=(\\mathbf{X},\\mathbf{F}", + "(\\mathbf{X},\\mathbf{G}_1(\\mathbf{X})))=(\\mathbf{X},\\mathbf{0})", + "$$", + "for all $\\mathbf{X}$ in $N$.", + "Since $\\boldsymbol{\\Phi}$ is one-to-one on $M$,", + "this implies that $\\mathbf{G}(\\mathbf{X})=", + "\\mathbf{G}_1(\\mathbf{X})$.", + "Since the partial derivatives", + "$$", + "\\frac{\\partial h_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le", + "n,", + "$$", + "are continuous functions of $(\\mathbf{X},\\mathbf{V})$ on $\\widehat{M}$, they", + "are continuous with respect to $\\mathbf{X}$ on the subset", + "$\\set{(\\mathbf{X},\\mathbf{0})}{\\mathbf{X} \\in N}$ of $\\widehat{M}$.", + "Therefore,", + "$\\mathbf{G}$ is", + "continuously differentiable on $N$. To verify \\eqref{eq:6.4.7}, we write", + "$\\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=\\mathbf{0}$ in terms of components;", + "thus,", + "$$", + "f_i(x_1,x_2, \\dots,x_n,g_1(\\mathbf{X}),g_2(\\mathbf{X}), \\dots,g_m(\\mathbf{X}))", + "=0,\\quad 1\\le i\\le m,\\quad\\mathbf{X}\\in N.", + "$$", + "Since $f_i$ and $g_1$, $g_2$, \\dots, $g_m$ are continuously", + "differentiable on their respective domains, the chain rule", + "(Theorem~\\ref{thmtype:5.4.3}) implies that", + "\\begin{equation} \\label{eq:6.4.11}", + "\\frac{\\partial f_i(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))}{\\partial x_j}+", + "\\sum^m_{r=1}", + "\\frac{\\partial f_i(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))}{\\partial u_r}", + "\\frac{\\partial g_r(\\mathbf{X})", + "}{\\partial x_j}=0,\\quad 1\\le i\\le m,\\ 1\\le j\\le n,", + "\\end{equation}", + "or, in matrix form,", + "\\begin{equation} \\label{eq:6.4.12}", + "\\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))+\\mathbf{F}_\\mathbf{U}", + "(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))\\mathbf{G}'(\\mathbf{X})=\\mathbf{0}.", + "\\end{equation}", + "Since $(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))\\in M$ for all $\\mathbf{X}$", + "in $N$ and $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$ is nonsingular when", + "$(\\mathbf{X},\\mathbf{U})\\in M$, we can multiply \\eqref{eq:6.4.12} on the left by", + "$\\mathbf{F}^{-1}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))$ to obtain", + "\\eqref{eq:6.4.7}. This completes the proof." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:5.4.3" + ], + "ref_ids": [ + 293, + 293, + 163 + ] + } + ], + "ref_ids": [] + }, + { + "id": 189, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.3", + "categories": [], + "title": "", + "contents": [ + "If $f$ is unbounded on the nondegenerate rectangle $R$ in", + "$\\R^n,$ then $f$ is not integrable on $R.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will show that if $f$ is unbounded on $R$, ${\\bf", + "P}=\\{R_1,R_2, \\dots,R_k\\}$ is", + "any partition of $R$, and $M>0$, then there are Riemann sums $\\sigma$", + "and $\\sigma'$ of $f$ over ${\\bf P}$ such that", + "\\begin{equation} \\label{eq:7.1.11}", + "|\\sigma-\\sigma'|\\ge M.", + "\\end{equation}", + "This implies that", + "$f$ cannot satisfy Definition~\\ref{thmtype:7.1.2}. (Why?)", + "Let", + "$$", + "\\sigma=\\sum_{j=1}^kf(\\mathbf{X}_j)V(R_j)", + "$$", + "be a Riemann sum of $f$ over ${\\bf P}$. There must be", + "an integer $i$ in $\\{1,2, \\dots,k\\}$ such that", + "\\begin{equation} \\label{eq:7.1.12}", + "|f(\\mathbf{X})-f(\\mathbf{X}_i)|\\ge\\frac{M }{ V(R_i)}", + "\\end{equation}", + "for some $\\mathbf{X}$ in $R_i$, because if this were not so, we", + "would have", + "$$", + "|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", + "\\quad 1\\le j\\le k.", + "$$", + "If this is so, then", + "\\begin{eqnarray*}", + "|f(\\mathbf{X})|\\ar=|f(\\mathbf{X}_j)+f(\\mathbf{X})-f(\\mathbf{X}_j)|\\le|f(\\mathbf{X}_j)|+|f(\\mathbf{X})-f(\\mathbf{X}_j)|\\\\", + "\\ar\\le |f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", + "1\\le j\\le k.", + "\\end{eqnarray*}", + "However, this implies that", + "$$", + "|f(\\mathbf{X})|\\le\\max\\set{|f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)}}{1\\le j\\le k},", + "\\quad \\mathbf{X}\\in R,", + "$$", + "which contradicts the assumption that $f$ is unbounded on $R$.", + " Now suppose that $\\mathbf{X}$ satisfies \\eqref{eq:7.1.12}, and", + "consider the Riemann sum", + "$$", + "\\sigma'=\\sum_{j=1}^nf(\\mathbf{X}_j')V(R_j)", + "$$", + "over the same partition ${\\bf P}$, where", + "$$", + "\\mathbf{X}_j'=\\left\\{\\casespace\\begin{array}{ll}", + "\\mathbf{X}_j,&j \\ne i,\\\\", + "\\mathbf{X},&j=i.\\end{array}\\right.", + "$$", + "Since", + "$$", + "|\\sigma-\\sigma'|=|f(\\mathbf{X})-f(\\mathbf{X}_i)|V(R_i),", + "$$", + "\\eqref{eq:7.1.12} implies \\eqref{eq:7.1.11}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.2" + ], + "ref_ids": [ + 359 + ] + } + ], + "ref_ids": [] + }, + { + "id": 190, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.5", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be bounded on a rectangle $R$ and let $\\mathbf{P}$", + "be a partition of $R.$ Then", + "\\begin{alist}", + "\\item % (a)", + " The upper sum $S(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the supremum", + "of the set of all Riemann sums of $f$ over $\\mathbf{P}.$", + "\\item % (b)", + " The lower sum $s(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the infimum", + " of the set of all Riemann sums of $f$ over $\\mathbf{P}.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.5}.", + "If", + "$$", + "m\\le f(\\mathbf{X})\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$},", + "$$", + "then", + "$$", + "mV(R)\\le s({\\bf P})\\le S({\\bf P})\\le MV(R);", + "$$", + "therefore, $\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}$ and", + "$\\underline{\\int_R}\\, f(\\mathbf{X})\\, d\\mathbf{X}$ exist, are unique, and", + "satisfy the inequalities", + "$$", + "mV(R)\\le\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le MV(R)", + "$$", + "and", + "$$", + "mV(R)\\le\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le MV(R).", + "$$", + "The upper and lower integrals are also written as", + "$$", + "\\overline{\\int_R}\\, f(x,y) \\,d(x,y)\\mbox{\\quad and\\quad}\\underline{\\int_R}\\,", + "f(x,y) \\,d(x,y)\\quad (n=2),", + "$$", + "$$", + "\\overline{\\int_R}\\, f(x,y,z) \\,d(x,y,z)\\mbox{\\quad and\\quad}", + "\\underline{\\int_R}\\, f(x,y,z) \\,d(x,y,z)\\quad (n=3),", + "$$", + "or", + "$$", + "\\overline{\\int_R}\\, f(x_1,x_2, \\dots,x_n) \\,d(x_1,x_2, \\dots,x_n)", + "$$", + "and", + "$$", + "\\underline{\\int_R}\\, f(x_1,x_2, \\dots,x_n)\\,d(x_1,x_2, \\dots,x_n)\\quad", + "\\mbox{\\quad ($n$ arbitrary)}.", + "$$", + "\\begin{example}\\label{example:7.1.2}\\rm", + "Find $\\underline{\\int_R}\\,f(x,y)\\,d(x,y)$ and", + " $\\overline{\\int_R}\\,f(x,y)\\,d(x,y)$, with", + "$R=[a,b]\\times [c,d]$ and", + "$$", + "f(x,y)=x+y,", + "$$", + "as in Example~\\ref{example:7.1.1}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 191, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.7", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on a rectangle $R,$ then", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}", + "\\le\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.8}.", + "The next theorem is analogous to Theorem~3.2.3." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 192, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on a rectangle $R,$ then", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X} =\\int_R f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.9}.", + "\\newpage", + "\\enlargethispage{\\baselineskip}", + "\\begin{lemma}\\label{thmtype:7.1.9}", + "If $f$ is bounded on a rectangle $R$ and $\\epsilon>0,$ there is", + " a $\\delta>0$ such that", + "\\vspace{4pt}", + "$$", + "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le S({\\bf P})<\\overline{\\int_R}\\,", + "f(\\mathbf{X})\\,d\\mathbf{X}+\\epsilon", + "$$", + "\\vspace{4pt}", + "and", + "\\vspace{4pt}", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\ge s({\\bf P})>", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}-\\epsilon", + "$$", + "\\vspace{4pt}", + "if $\\|{\\bf P}\\|<\\delta.$", + "\\end{lemma}" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 193, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.10", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on a rectangle $R$ and", + "\\vspace{2pt}", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=L,", + "$$", + "\\vspace{2pt}", + "then $f$ is integrable on $R,$ and", + "\\vspace{2pt}", + "$$", + "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.11}.", + "Theorems~\\ref{thmtype:7.1.8} and \\ref{thmtype:7.1.10}", + " imply the following theorem, which is analogous to", + "Theorem~\\ref{thmtype:3.2.6}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.8", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.10", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.6" + ], + "ref_ids": [ + 192, + 193, + 49 + ] + } + ], + "ref_ids": [] + }, + { + "id": 194, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.11", + "categories": [], + "title": "", + "contents": [ + "A bounded", + "function $f$ is integrable on a rectangle $R$ if and only if", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=\\overline{\\int_R}\\, f(\\mathbf{X})\\,", + "d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 195, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.12", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on a rectangle $R,$ then $f$ is integrable on $R$", + "if and only if for every $\\epsilon>0$ there is a partition ${\\bf P}$", + "of $R$ such that", + "$$", + "S({\\bf P})-s({\\bf P})<\\epsilon.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.12}.", + "Theorem~\\ref{thmtype:7.1.12} provides a useful criterion for", + "integrability. The next theorem is an important application.", + "It is analogous to", + "Theorem~\\ref{thmtype:3.2.8}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.12", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.8" + ], + "ref_ids": [ + 195, + 51 + ] + } + ], + "ref_ids": [] + }, + { + "id": 196, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.13", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a rectangle $R$ in $\\R^n,$ then $f$ is", + "integrable on~$R.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $\\epsilon>0$. Since $f$ is uniformly continuous on $R$", + "(Theorem~\\ref{thmtype:5.2.14}), there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:7.1.23}", + "|f(\\mathbf{X})-f(\\mathbf{X}')|<\\frac{\\epsilon}{ V({\\bf R})}", + "\\end{equation}", + "if $\\mathbf{X}$ and $\\mathbf{X}'$ are in $R$ and", + " $|\\mathbf{X}-\\mathbf{X}'|<\\delta$. Let ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ be a partition of", + "$R$ with $\\|P\\|<\\delta/\\sqrt n$. Since $f$ is continuous on $R$, there", + "are points $\\mathbf{X}_j$ and $\\mathbf{X}_j'$ in $R_j$ such that", + "$$", + "f(\\mathbf{X}_j)=M_j=\\sup_{\\mathbf{X}\\in R_j}f(\\mathbf{X})", + "\\mbox{\\quad and \\quad}", + "f(\\mathbf{X}_j')=m_j=\\inf_{\\mathbf{X}\\in R_j}f(\\mathbf{X})", + "$$", + "(Theorem~\\ref{thmtype:5.2.12}).", + "Therefore,", + "$$", + "S(\\mathbf{P})-s(\\mathbf{P})=\\sum_{j=1}^n(f(\\mathbf{X}_j)-", + "f(\\mathbf{X}_j'))V(R_j).", + "$$", + "Since $\\|{\\bf P}\\|<\\delta/\\sqrt n$,", + "$|\\mathbf{X}_j-\\mathbf{X}_j'|<\\delta$, and, from \\eqref{eq:7.1.23}", + "with $\\mathbf{X}=\\mathbf{X}_j$ and $\\mathbf{X}'=\\mathbf{X}_j'$,", + "$$", + " S(\\mathbf{P})-s(\\mathbf{P})<\\frac{\\epsilon}{ V(R)}", + "\\sum_{j=1}^kV(R_j)=\\epsilon.", + "$$", + "Hence, $f$ is integrable", + "on $R$, by Theorem~\\ref{thmtype:7.1.12}.", + "\\boxit{Sets with Zero Content}", + "The next definition will enable us to establish the existence", + "of $\\int_Rf(\\mathbf{X})\\,d\\mathbf{X}$ in cases where $f$ is bounded on the", + "rectangle $R$, but is not necessarily continuous for all $\\mathbf{X}$", + "in $R$.", + "\\begin{definition}\\label{thmtype:7.1.14}", + "A subset $E$ of $\\R^n$ has zero content if for each", + "$\\epsilon>0$", + "there is a finite set of rectangles $T_1$, $T_2$, \\dots, $T_m$ such", + "that", + "\\begin{equation}\\label{eq:7.1.24}", + "E\\subset\\bigcup_{j=1}^m T_j", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.1.25}", + "\\sum_{j=1}^m V(T_j)<\\epsilon.", + "\\end{equation}", + "\\end{definition}", + "\\begin{example}\\label{example:7.1.3}\\rm Since the empty set is contained", + "in every rectangle, the empty set has zero content. If $E$ consists of", + "finitely", + "many points $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots,", + "$\\mathbf{X}_m$, then $\\mathbf{X}_j$ can be enclosed in a rectangle $T_j$", + "such that", + "$$", + "V(T_j)<\\frac{\\epsilon}{ m},\\quad 1\\le j\\le m.", + "$$", + "Then \\eqref{eq:7.1.24} and \\eqref{eq:7.1.25} hold, so $E$ has zero content.", + "\\end{example}", + "\\begin{example}\\label{example:7.1.4}\\rm Any bounded set $E$ with only", + "finitely many limit points has zero content. To see this, we first", + "observe that if $E$ has no limit points, then it must be finite, by", + "the Bolzano--Weierstrass theorem (Theorem~\\ref{thmtype:1.3.8}), and", + "therefore must have zero content,", + "by Example~\\ref{example:7.1.3}. Now suppose that the limit points of $E$ are", + "$\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots, $\\mathbf{X}_m$. Let $R_1$, $R_2$,", + "\\dots, $R_m$ be rectangles such that", + "$\\mathbf{X}_i\\in R^0_i$ and", + "\\begin{equation}\\label{eq:7.1.26}", + "V(R_i)<\\frac{\\epsilon}{2m},\\quad 1\\le i\\le m.", + "\\end{equation}", + "The set of points of $E$ that are not in $\\cup_{j=1}^mR_j$ has no", + "limit points (why?) and, being bounded, must be finite (again by the", + "Bolzano--Weierstrass theorem). If this set contains $p$ points,", + "then it can be covered by rectangles", + "$R_1'$, $R_2'$, \\dots, $R_p'$ with", + "\\begin{equation}\\label{eq:7.1.27}", + "V(R_j')<\\frac{\\epsilon}{2p},\\quad 1\\le j\\le p.", + "\\end{equation}", + "Now,", + "$$", + "E\\subset\\left(\\bigcup_{i=1}^mR_i\\right)\\bigcup\\left(\\bigcup^p_{j=1}", + "R_j'\\right)", + "$$", + "and, from \\eqref{eq:7.1.26} and \\eqref{eq:7.1.27},", + "$$", + "\\sum_{i=1}^m V(R_i)+\\sum_{j=1}^p V(R_j')<\\epsilon.", + "$$", + "\\end{example}", + "\\begin{example}\\label{example:7.1.5}\\rm", + " If $f$ is continuous on $[a,b]$,", + "then the curve", + "\\begin{equation}\\label{eq:7.1.28}", + "y=f(x),\\quad a\\le x\\le b", + "\\end{equation}", + "(that is, the set $\\set{(x,y)}{y=f(x),\\ a\\le x\\le b})$, has zero", + "content in $\\R^2$. To see this, suppose that $\\epsilon>0$, and", + "choose $\\delta>0$ such that", + "\\begin{equation}\\label{eq:7.1.29}", + "|f(x)-f(x')|<\\epsilon\\mbox{\\quad if\\quad} x, x'\\in [a,b]", + "\\mbox{\\quad and\\quad} |x-x'|<\\delta.", + "\\end{equation}", + "This is possible because $f$ is uniformly continuous on $[a,b]$", + "(Theorem~\\ref{thmtype:2.2.12}). Let", + "$$", + "P: a=x_00$. Since $E$ has zero content, there are", + "rectangles", + "$T_1$, $T_2$, \\dots, $T_m$ such that", + "\\begin{equation} \\label{eq:7.1.31}", + "E\\subset\\bigcup_{j=1}^m T_j", + "\\end{equation}", + "and", + "\\begin{equation} \\label{eq:7.1.32}", + "\\sum_{j=1}^m V(T_j)<\\epsilon.", + "\\end{equation}", + " We may assume that", + "$T_1$, $T_2$, \\dots, $T_m$ are contained in $R$, since, if not, their", + "intersections with", + "$R$ would be contained in $R$, and still satisfy \\eqref{eq:7.1.31}", + "and \\eqref{eq:7.1.32}.", + " We may also assume that if $T$ is any rectangle such", + "that", + "\\begin{equation}\\label{eq:7.1.33}", + "T\\bigcap\\left(\\bigcup_{j=1}^m T_j^0\\right)=\\emptyset, \\mbox{\\quad", + "then", + "\\quad}T\\cap E=\\emptyset", + "\\end{equation}", + "\\newpage", + "\\noindent", + "since if this were not so, we could make it so by enlarging", + "$T_1$, $T_2$, \\dots, $T_m$", + "slightly while maintaining \\eqref{eq:7.1.32}. Now suppose that", + "\\vspace*{1pt}", + "$$", + "T_j=[a_{1j},b_{1j}]\\times [a_{2j},b_{2j}]\\times\\cdots\\times", + "[a_{nj},b_{nj}],\\quad 1\\le j\\le m,", + "$$", + "\\vspace*{1pt}", + "\\noindent let $P_{i0}$ be the partition of $[a_i,b_i]$ (see", + "\\eqref{eq:7.1.30}) with partition points", + "$$", + "a_i,b_i,a_{i1},b_{i1},a_{i2},b_{i2}, \\dots,a_{im},b_{im}", + "\\vspace*{1pt}", + "$$", + "(these are not in increasing order), $1\\le i\\le n$, and let", + "\\vspace*{1pt}", + "$$", + "{\\bf P}_0=P_{10}\\times P_{20}\\times\\cdots\\times P_{n0}.", + "$$", + "\\vspace*{1pt}", + "\\noindent\\hskip-.3em Then ${\\bf P}_0$ consists of rectangles whose", + "union equals $\\cup_{j=1}^m T_j$", + "and other rectangles", + "$T'_1$, $T'_2$, \\dots, $T'_k$ that do not intersect $E$. (We need", + "\\eqref{eq:7.1.33} to be sure that $T'_i\\cap E=\\emptyset,", + "1\\le i\\le k.)$ If we let", + "$$", + "B=\\bigcup_{j=1}^m T_j\\mbox{\\quad and\\quad} C=\\bigcup^k_{i=1} T'_i,", + "$$", + "then $R=B\\cup C$ and $f$ is continuous on the compact set $C$.", + "If ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a refinement of ${\\bf P}_0$,", + "then every subrectangle $R_j$ of ${\\bf P}$ is contained entirely in", + "$B$ or entirely in $C$. Therefore, we can write", + "\\vspace*{1pt}", + "\\begin{equation}\\label{eq:7.1.34}", + "S({\\bf P})-s({\\bf P})=\\Sigma_1(M_j-m_j)", + "V(R_j)+\\Sigma_2(M_j-m_j)V(R_j),", + "\\end{equation}", + "\\vspace*{1pt}", + "\\noindent \\hskip-.3em", + "where $\\Sigma_1$ and $\\Sigma_2$ are summations over values of $j$ for", + "which $R_j\\subset B$ and $R_j\\subset C$, respectively. Now suppose that", + "$$", + "|f(\\mathbf{X})|\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$}.", + "$$", + "Then", + "\\begin{equation}\\label{eq:7.1.35}", + "\\Sigma_1(M_j-m_j) V(R_j)\\le2M\\,\\Sigma_1 V(R_j)=2M\\sum_{j=1}^m V(T_j)<", + "2M\\epsilon,", + "\\end{equation}", + "from \\eqref{eq:7.1.32}.", + "Since $f$ is uniformly continuous on the compact set $C$", + "(Theorem~\\ref{thmtype:5.2.14}),", + "there is a $\\delta>0$ such that $M_j-m_j<\\epsilon$ if", + "$\\|{\\bf P}\\|< \\delta$ and $R_j\\subset C$; hence,", + "$$", + "\\Sigma_2(M_j-m_j)V(R_j)<\\epsilon\\Sigma_2\\, V(R_j)\\le\\epsilon V(R).", + "$$", + "This, \\eqref{eq:7.1.34}, and \\eqref{eq:7.1.35} imply that", + "$$", + "S({\\bf P})-s({\\bf P})<[2M+V(R)]\\epsilon", + "$$", + "if $\\|{\\bf P}\\|<\\delta$ and ${\\bf P}$ is a refinement of ${\\bf P}_0$.", + "Therefore, Theorem~\\ref{thmtype:7.1.12} implies that $f$ is integrable on", + "$R$.", + "\\enlargethispage{4\\baselineskip}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" + ], + "ref_ids": [ + 154, + 195 + ] + } + ], + "ref_ids": [] + }, + { + "id": 198, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.19", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is bounded on a bounded set $S$ and continuous", + "except on a subset $E$ of $S$ with zero content. Suppose also that", + "$\\partial S$ has zero content$.$ Then $f$ is integrable on $S.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $f_S$ be as in \\eqref{eq:7.1.36}. Since a discontinuity of", + "$f_S$ is either a discontinuity of $f$ or a point of $\\partial S$, the", + "set of discontinuities of $f_S$ is the union of two sets of zero", + "content and therefore is of zero content (Lemma~\\ref{thmtype:7.1.15}).", + "Therefore, $f_S$ is integrable on any rectangle containing $S$", + "(from Theorem~\\ref{thmtype:7.1.16}), and consequently on $S$", + "(Definition~\\ref{thmtype:7.1.17})." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.15", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.16", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.17" + ], + "ref_ids": [ + 261, + 197, + 362 + ] + } + ], + "ref_ids": [] + }, + { + "id": 199, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.21", + "categories": [], + "title": "", + "contents": [ + "A differentiable surface in $\\R^n$ has zero content$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $S$, $D$, and $\\mathbf{G}$ be as in Definition~\\ref{thmtype:7.1.20}.", + "From Lemma~\\ref{thmtype:6.2.7}, there is a constant $M$ such", + "that", + "\\begin{equation}\\label{eq:7.1.37}", + "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|\\le", + "M|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D.", + "\\end{equation}", + "Since $D$ is bounded, $D$ is contained in a cube", + "$$", + "C=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_m,b_m],", + "$$", + "where", + "$$", + "b_i-a_i=L,\\quad 1\\le i\\le m.", + "$$", + "Suppose that we partition $C$ into $N^m$ smaller cubes by partitioning", + "each of the intervals $[a_i,b_i]$ into $N$ equal subintervals. Let", + "$R_1$, $R_2$, \\dots, $R_k$ be the smaller cubes so produced that", + "contain", + "points of $D$, and select points $\\mathbf{X}_1$, $\\mathbf{X}_2$, \\dots,", + "$\\mathbf{X}_k$", + "such that $\\mathbf{X}_i\\in D\\cap R_i$, $1\\le i\\le k$. If $\\mathbf{Y}", + "\\in D\\cap R_i$, then \\eqref{eq:7.1.37} implies that", + "\\begin{equation}\\label{eq:7.1.38}", + "|\\mathbf{G}(\\mathbf{X}_i)-\\mathbf{G}(\\mathbf{Y})|\\le M|\\mathbf{X}_i-\\mathbf{Y}|.", + "\\end{equation}", + "Since $\\mathbf{X}_i$ and $\\mathbf{Y}$ are both in the cube $R_i$ with", + "edge length $L/N$,", + "$$", + "|\\mathbf{X}_i-\\mathbf{Y}|\\le\\frac{L\\sqrt{m}}{ N}.", + "$$", + " This and \\eqref{eq:7.1.38} imply that", + "$$", + "|\\mathbf{G}(\\mathbf{X}_i)-\\mathbf{G}(\\mathbf{Y})|\\le\\frac{ML\\sqrt m}{ N},", + "$$", + "which in turn implies that", + "$\\mathbf{G}(\\mathbf{Y})$ lies in a cube $\\widetilde{R}_i$ in $\\R^n$", + " centered at $\\mathbf{G}(\\mathbf{X}_i)$,", + "with", + "sides of length $2ML\\sqrt{m}/N$.", + " Now", + "$$", + "\\sum_{i=1}^k V(\\widetilde{R}_i)= k\\left(\\frac{2ML\\sqrt{m}}{", + "N}\\right)^n\\le", + "N^m\\left(\\frac{2ML\\sqrt{m}}{ N}\\right)^n=(2ML\\sqrt{m})^n", + "N^{m-n}.", + "$$", + "Since $n>m$, we can make the sum on the left arbitrarily small by", + "taking $N$ sufficiently large. Therefore, $S$ has zero content." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.20", + "TRENCH_REAL_ANALYSIS-thmtype:6.2.7" + ], + "ref_ids": [ + 364, + 258 + ] + } + ], + "ref_ids": [] + }, + { + "id": 200, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.22", + "categories": [], + "title": "", + "contents": [ + "Suppose that $S$ is a bounded set in $\\R^n,$ with boundary", + "consisting of a finite number of differentiable surfaces$.$ Let $f$ be", + "bounded on $S$ and continuous except on a set of zero content. Then", + "$f$ is integrable on $S.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 201, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.23", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are integrable on $S,$ then so is $f+g,$ and", + "$$", + "\\int_S(f+g)(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}+", + "\\int_S g(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.20}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 202, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.24", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on $S$ and $c$ is a constant$,$ then $cf$ is", + "integrable on $S,$ and", + "$$", + "\\int_S(cf)(\\mathbf{X})\\,d\\mathbf{X}=c\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.21}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 203, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.25", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are integrable on $S$ and $f(\\mathbf{X})\\le g(\\mathbf{X})$", + "for $\\mathbf{X}$ in $S,$ then", + "$$", + "\\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\le\\int_S g(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.22}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 204, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.26", + "categories": [], + "title": "", + "contents": [ + " If $f$ is integrable on $S,$", + "then so is $|f|,$ and", + "$$", + "\\left|\\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le\\int_S |f(\\mathbf{X})|\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.23}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 205, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.27", + "categories": [], + "title": "", + "contents": [ + "If $f$ and $g$ are integrable on $S,$ then so is the product $fg.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.24}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 206, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.28", + "categories": [], + "title": "", + "contents": [ + "Suppose that $u$ is continuous and $v$ is integrable and nonnegative", + "on a rectangle $R.$ Then", + "$$", + "\\int_R u(\\mathbf{X})v(\\mathbf{X})\\,d\\mathbf{X}=", + "u(\\mathbf{X}_0)\\int_R v(\\mathbf{X})\\,d\\mathbf{X}", + "$$", + "for some $\\mathbf{X}_0$ in $R.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.25}.", + "\\begin{lemma}\\label{thmtype:7.1.29}", + "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable", + "on $S.$ Then", + " $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and", + "$$", + "\\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$", + "\\end{lemma}", + "\\nopagebreak" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 207, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.30", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on disjoint sets $S_1$ and $S_2,$ then $f$ is", + "integrable on $S_1\\cup S_2,$ and", + "\\begin{equation}\\label{eq:7.1.39}", + "\\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+", + "\\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For $i=1$, $2$, let", + "$$", + "f_{S_i}(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f(\\mathbf{X}),&\\mathbf{X}\\in", + "S_i,\\\\[2\\jot]", + " 0,&\\mathbf{X}\\not\\in S_i.\\end{array}\\right.", + "$$", + "From Lemma~\\ref{thmtype:7.1.29} with $S=S_i$ and $T=S_1\\cup S_2$,", + "$f_{S_i}$ is integrable on $S_1\\cup S_2$, and", + "$$", + "\\int_{S_1\\cup S_2} f_{S_i}(\\mathbf{X})\\,d\\mathbf{X}", + "=\\int_{S_i} f(\\mathbf{X})\\,d\\mathbf{X},\\quad i=1,2.", + "$$", + "Theorem~\\ref{thmtype:7.1.23} now implies that $f_{S_1}+f_{S_2}$ is integrable on", + "$S_1\\cup S_2$ and", + "\\begin{equation}\\label{eq:7.1.40}", + "\\int_{S_1\\cup S_2} (f_{S_1}+f_{S_2})(\\mathbf{X})\\,d\\mathbf{X}=\\int_{S_1}", + "f(\\mathbf{X})\\,d\\mathbf{X}+\\int_{S_2} f(\\mathbf{X})\\, d\\mathbf{X}.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Since $S_1\\cap S_2=\\emptyset$,", + "$$", + "\\left(f_{S_1}+f_{S_2}\\right)(\\mathbf{X})=", + "f_{S_1}(\\mathbf{X})+f_{S_2}(\\mathbf{X})", + "=f(\\mathbf{X}),\\quad \\mathbf{X}\\in S_1\\cup S_2.", + "$$", + " Therefore,", + "\\eqref{eq:7.1.40} implies \\eqref{eq:7.1.39}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.29", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.23" + ], + "ref_ids": [ + 262, + 201 + ] + } + ], + "ref_ids": [] + }, + { + "id": 208, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.1", + "categories": [], + "title": "", + "contents": [ + "$R= [a,b]\\times [c,d]$ and", + "$$", + " F(y)=\\int_a^b f(x,y)\\,dx", + "$$", + "exists for each $y$ in $[c,d].$ Then $F$ is integrable on $[c,d],$", + "and", + "\\begin{equation}\\label{eq:7.2.1}", + "\\int_c^d F(y)\\,dy=\\int_R f(x,y)\\,d(x,y);", + "\\end{equation}", + "that is$,$", + "\\begin{equation}\\label{eq:7.2.2}", + "\\int_c^d dy\\int_a^b f(x,y)\\,dx=\\int_R f(x,y)\\,d(x,y).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$$", + "P_1: a=x_00$ a", + "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", + "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from", + "\\eqref{eq:7.2.6}, there is", + "a partition $P_2$ of $[c,d]$ such that", + "$S_F(P_2)-s_F(P_2)<\\epsilon$,", + " so $F$ is integrable on $[c,d]$, from", + "Theorem~\\ref{thmtype:3.2.7}.", + "It remains to verify \\eqref{eq:7.2.1}. From \\eqref{eq:7.2.4} and the", + "definition of $\\int_c^dF(y)\\,dy$,", + "there is for each $\\epsilon>0$ a $\\delta>0$ such that", + "$$", + "\\left|\\int_c^d F(y)\\,dy-\\sigma\\right|<\\epsilon\\mbox{\\quad if\\quad}", + "\\|P_2\\|<\\delta;", + "$$", + "that is,", + "$$", + "\\sigma-\\epsilon<\\int_c^d F(y)\\,dy<\\sigma+\\epsilon\\mbox{\\quad if \\quad}", + "\\|P_2\\|<\\delta.", + "$$", + "This and \\eqref{eq:7.2.5} imply that", + "$$", + "s_f(\\mathbf{P})-\\epsilon<\\int_c^d F(y)\\,dy0$ a", + "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", + "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from \\eqref{eq:7.2.11},", + "there", + "is a partition $\\mathbf{Q}$ of $T$ such that", + "$S_{F_p}(\\mathbf{Q})-s_{F_p}(\\mathbf{Q})<\\epsilon$, so $F_p$ is integrable", + "on $T$, from Theorem~\\ref{thmtype:7.1.12}.", + "It remains to verify that", + "\\begin{equation} \\label{eq:7.2.12}", + "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}.", + "\\end{equation}", + "From \\eqref{eq:7.2.9} and the definition of $\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}$, there", + "is for each $\\epsilon>0$ a $\\delta>0$ such that", + "$$", + "\\left|\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "-\\sigma\\right|<\\epsilon\\mbox{\\quad", + "if\\quad}", + "\\|\\mathbf{Q}\\|<\\delta;", + "$$", + "that is,", + "$$", + "\\sigma-\\epsilon<\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "<\\sigma+", + "\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{Q}\\|<\\delta.", + "$$", + "This and \\eqref{eq:7.2.10} imply that", + "$$", + "s_f(\\mathbf{P})-\\epsilon<", + "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "2$ and the proposition is true with $n$ replaced", + "by $n-1$. Holding $x_n$ fixed and applying this assumption", + "yields", + "$$", + "F_n(x_n)=", + "\\int^{b_{n-1}}_{a_{n-1}}", + "dx_{n-1}\\int_{a_{n-2}}^{b_{n-2}}dx_{n-2}\\cdots", + "\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} f(\\mathbf{X})\\,dx_1.", + "$$", + "Now Theorem~\\ref{thmtype:7.2.3} with $p=n-1$ completes the induction." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:7.2.3" + ], + "ref_ids": [ + 208, + 209 + ] + } + ], + "ref_ids": [] + }, + { + "id": 211, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.5", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on", + "$$", + "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n],", + "$$", + "then $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}$ can be evaluated by iterated", + "integrals in any of the $n!$ ways indicated in $\\eqref{eq:7.2.16}.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 212, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.6", + "categories": [], + "title": "", + "contents": [ + "If $f$ is integrable on the set $S$ in $\\eqref{eq:7.2.17}$ and the", + "integral $\\eqref{eq:7.2.19}$ exists for $c\\le y\\le d,$ then", + "\\begin{equation}\\label{eq:7.2.20}", + "\\int_S f(x,y) \\,d(x,y)=\\int_c^d dy\\int^{v(y)}_{u(y)} f(x,y)\\,dx.", + "\\end{equation}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 213, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.2.7", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is integrable on", + "$$", + "S=\\set{(x,y,z)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z),\\", + "c\\le z\\le d},", + "$$", + "and let", + "$$", + "S(z)=\\set{(x,y)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z)}", + "$$", + "for each $z$ in $[c,d].$ Then", + "$$", + "\\int_S f(x,y,z)\\,d(x,y,z)=\\int_c^d dz\\int^{v_2(z)}_{u_2(z)} dy", + "\\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx,", + "$$", + "provided that", + "$$", + "\\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx", + "$$", + "exists for all $(y,z)$ such that", + "$$", + "c\\le z\\le d\\mbox{\\quad and\\quad} u_2(z)\\le y\\le v_2(z),", + "$$", + "and", + "$$", + "\\int_{S(z)} f(x,y,z)\\,d(x,y)", + "$$", + "exists for all $z$ in $[c,d].$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 214, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", + "categories": [], + "title": "", + "contents": [ + "A bounded set $S$ is Jordan measurable if and only if the boundary", + "of $S$ has", + "zero content$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $R$ be a rectangle containing $S$. Suppose that $V(\\partial S)=0$.", + "Since", + "$\\psi_{S}$ is bounded on $R$ and discontinuous only on", + "$\\partial S$", + "(Exercise~\\ref{exer:2.2.9}), Theorem~\\ref{thmtype:7.1.19}", + "implies that $\\int_R\\psi_S (\\mathbf{X})\\,d\\mathbf{X}$ exists.", + " For the converse, suppose that", + "$\\partial S$ does not have zero content", + "and let ${\\bf P}=\\{R_1, R_2,\\dots, R_k\\}$ be a partition", + "of $R$. For each $j$ in $\\{1,2,\\dots,k\\}$ there are three", + "possibilities:", + "\\begin{description}", + " \\item{1.} $R_j\\subset S$; then", + "$$", + "\\min\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=", + "\\max\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=1.", + "$$", + "\\item{2.} $R_j\\cap S\\ne\\emptyset$ and $R_j\\cap S^c\\ne", + "\\emptyset$; then", + "$$", + "\\min\\set{\\psi_S (\\mathbf{X})}{\\mathbf{X}\\in R_j}=0\\mbox{\\quad and\\quad}", + "\\max\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=1.", + "$$", + "\\item{3.} $R_j\\subset S^c$; then", + "$$", + "\\min\\set{\\psi_S(\\mathbf{X})}{\\mathbf{X}\\in R_j}=\\max\\set{\\psi_S(\\mathbf{X})}", + "{\\mathbf{X}\\in R_j}=0.", + "$$", + "\\end{description}", + "\\newpage", + "\\noindent Let", + "\\begin{equation} \\label{eq:7.3.2}", + "{\\mathcal U}_1=\\set{j}{R_j\\subset S}", + "\\mbox{\\quad and \\quad}", + "{\\mathcal U}_2=\\set{j}{R_j\\cap S\\ne\\emptyset\\mbox{ and }R_j\\cap", + "S^c\\ne\\emptyset}.", + "\\end{equation}", + "Then the upper and lower", + "sums of $\\psi_S$ over ${\\bf P}$ are", + "\\begin{equation}\\label{eq:7.3.3}", + "\\begin{array}{rcl}", + "S({\\bf P})\\ar=\\dst\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", + "V(R_j)\\\\[2\\jot]", + "\\ar=\\mbox{total content of the subrectangles in ${\\bf P}$ that intersect", + "$S$}", + "\\end{array}", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.3.4}", + "\\begin{array}{rcl}", + "s({\\bf P})\\ar=\\dst\\sum_{j\\in{\\mathcal U}_1} V(R_j) \\\\", + "\\ar=\\mbox{total content of the subrectangles in ${\\bf P}$", + "contained in $S$}.", + "\\end{array}", + "\\end{equation}", + "Therefore,", + "$$", + "S({\\bf P})-s({\\bf P})=\\sum_{j\\in {\\mathcal U}_2} V(R_j),", + "$$", + "which is the total content of the subrectangles in ${\\bf P}$ that", + "intersect both $S$ and $S^c$.", + " Since these subrectangles contain", + "$\\partial S$,", + "which does not have zero content, there is an", + "$\\epsilon_0>0$ such that", + "$$", + "S({\\bf P})-s({\\bf P})\\ge\\epsilon_0", + "$$", + "for every partition ${\\bf P}$ of $R$. By", + "Theorem~\\ref{thmtype:7.1.12}, this implies that $\\psi_S$ is not", + "integrable on $R$, so $S$ is not Jordan measurable." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.19", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" + ], + "ref_ids": [ + 198, + 195 + ] + } + ], + "ref_ids": [] + }, + { + "id": 215, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}:\\R^n\\to \\R^n$ is regular on a compact", + "Jordan measurable set $S.$ Then $\\mathbf{G}(S)$ is compact and", + "Jordan measurable$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We leave it to you to prove that $\\mathbf{G}(S)$ is", + "compact", + "(Exercise~6.2.23). Since $S$ is", + "Jordan measurable,", + " $V(\\partial S)=0$, by Theorem~\\ref{thmtype:7.3.1}.", + "Therefore, $V(\\mathbf{G}(\\partial S))=0$, by Lemma~\\ref{thmtype:7.3.4}.", + "But $\\mathbf{G}(\\partial S)=", + "\\partial(\\mathbf{G}(S))$ (Exercise~\\ref{exer:6.3.23}), so", + "$V(\\partial(\\mathbf{G}(S)))=0$, which implies that", + "$\\mathbf{G}(S)$ is Jordan measurable, again by Theorem~\\ref{thmtype:7.3.1}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.4", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.1" + ], + "ref_ids": [ + 214, + 264, + 214 + ] + } + ], + "ref_ids": [] + }, + { + "id": 216, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", + "categories": [], + "title": "", + "contents": [ + "If $S$ is a compact Jordan measurable subset", + " of $\\R^n$ and $\\mathbf{L}:\\R^n\\to \\R^n$ is the invertible linear", + "transformation", + "$\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{AY},$ then", + "\\begin{equation}\\label{eq:7.3.14}", + "V(\\mathbf{L}(S))=|\\det(\\mathbf{A})| V(S).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Theorem~\\ref{thmtype:7.3.5} implies that $\\mathbf{L}(S)$ is", + "Jordan measurable. If", + "\\begin{equation} \\label{eq:7.3.15}", + "V(\\mathbf{L}(R))=|\\det(\\mathbf{A})| V(R)", + "\\end{equation}", + "whenever $R$ is a rectangle, then", + " \\eqref{eq:7.3.14} holds if $S$", + "is any compact Jordan measurable set. To see this, suppose that", + "$\\epsilon>0$, let", + "$R$ be a rectangle containing $S$, and let", + "${\\bf P}=\\{R_1,R_2,\\dots,R_k\\}$ be a partition of $R$ such that the", + "upper and lower sums of $\\psi_S$ over ${\\bf", + "P}$ satisfy the inequality", + "\\begin{equation}\\label{eq:7.3.16}", + "S({\\bf P})-s({\\bf P})<\\epsilon.", + "\\end{equation}", + "Let ${\\mathcal U}_1$ and ${\\mathcal U}_2$ be as in \\eqref{eq:7.3.2}.", + "From \\eqref{eq:7.3.3} and \\eqref{eq:7.3.4},", + "\\begin{equation}\\label{eq:7.3.17}", + "s({\\bf P})=\\sum_{j\\in{\\mathcal U}_1} V(R_j)\\le V(S)\\le\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", + "V(R_j)=S({\\bf P}).", + "\\end{equation}", + " Theorem~\\ref{thmtype:7.3.7}", + "implies that $\\mathbf{L}(R_1)$, $\\mathbf{L}(R_2)$, \\dots, $\\mathbf{L}(R_k)$", + "and", + "$\\mathbf{L}(S)$ are all Jordan measurable.", + "Since", + "$$", + "\\bigcup_{j\\in{\\mathcal U}_1}R_j\\subset S\\subset\\bigcup_{j\\in{\\mathcal", + "S}_1\\cup{\\mathcal S_2}}R_j,", + "$$", + "it follows that", + "$$", + "L\\left(\\bigcup_{j\\in{\\mathcal U}_1}R_j\\right)\\subset", + "L(S)\\subset L\\left(\\bigcup_{j\\in{\\mathcal S}_1\\cup{\\mathcal S_2}}R_j\\right).", + "$$", + "Since $L$ is one-to-one on $\\R^n$, this implies that", + "\\begin{equation} \\label{eq:7.3.18}", + "\\sum_{j\\in{\\mathcal U}_1} V(\\mathbf{L}(R_j))\\le V(\\mathbf{L}(S))\\le\\sum_{j\\in{\\mathcal U}_1}", + "V(\\mathbf{L}(R_j))+\\sum_{j\\in{\\mathcal U}_2} V(\\mathbf{L}(R_j)).", + "\\end{equation}", + "If we assume that \\eqref{eq:7.3.15} holds whenever $R$ is a rectangle,", + "then", + "$$", + "V(\\mathbf{L}(R_j))=|\\det(\\mathbf{A})|V(R_j),\\quad 1\\le j\\le k,", + "$$", + "so \\eqref{eq:7.3.18} implies that", + "$$", + "s({\\bf P})\\le \\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\le S({\\bf P}).", + "$$", + "This, \\eqref{eq:7.3.16} and \\eqref{eq:7.3.17} imply that", + "$$", + "\\left|V(S)-\\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\right|<\\epsilon;", + "$$", + "hence, since $\\epsilon$ can be made arbitrarily small, \\eqref{eq:7.3.14}", + "follows for any Jordan measurable set.", + "To complete the proof, we must verify \\eqref{eq:7.3.15} for every", + "rectangle", + "$$", + "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]=I_1\\times", + "I_2\\times\\cdots\\times I_n.", + "$$", + " Suppose that $\\mathbf{A}$ in \\eqref{eq:7.3.12} is an elementary matrix;", + "that is, let", + "$$", + "\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{EY}.", + "$$", + "{\\sc Case 1}. If $\\mathbf{E}$ is obtained by interchanging the $i$th and", + "$j$th rows of $\\mathbf{I}$, then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$ and $r\\ne j$};\\\\", + "y_j&\\mbox{if $r=i$};\\\\", + "y_i&\\mbox{if $r=j$}.\\end{array}\\right.", + "$$", + "Then $\\mathbf{L}(R)$ is the Cartesian product of $I_1$,", + "$I_2$, \\dots, $I_n$ with", + "$I_i$ and $I_j$ interchanged, so", + "$$", + "V(\\mathbf{L}(R))=V(R)=|\\det(\\mathbf{E})|V(R)", + "$$", + "since $\\det(\\mathbf{E})=-1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "{\\sc Case 2}. If $\\mathbf{E}$ is obtained by multiplying the $r$th row of", + "$\\mathbf{I}$ by $a$, then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$},\\\\", + "ay_i&\\mbox{if $r=i$}.\\end{array}\\right.", + "$$", + "Then", + "$$", + "\\mathbf{L}(R)=I_1\\times\\cdots\\times I_{i-1}\\times I'_i\\times I_{i+1}\\times", + "\\cdots\\times I_n,", + "$$", + "where $I'_i$ is an interval with length equal to $|a|$ times the", + "length of $I_i$, so", + "$$", + "V(\\mathbf{L}(R))=|a|V(R)=|\\det(\\mathbf{E})|V(R)", + "$$", + "since $\\det(\\mathbf{E})=a$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "{\\sc Case 3}. If $\\mathbf{E}$ is obtained by adding $a$ times the $j$th", + "row of $\\mathbf{I}$ to its $i$th row ($j\\ne i$), then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$};\\\\", + "y_i+ay_j&\\mbox{if $r=i$}.\\end{array}\\right.", + "$$", + "Then", + "$$", + "\\mathbf{L}(R)=\\set{(x_1,x_2,\\dots,x_n)}{a_i+ax_j\\le x_i\\le b_i+ax_j", + "\\mbox{ and } a_r\\le x_r\\le b_r\\mbox{if } r\\ne i},", + "$$", + "which is a parallelogram if $n=2$ and a parallelepiped if $n=3$", + "(Figure~\\ref{figure:7.3.1}). Now", + "$$", + "V(\\mathbf{L}(R))=\\int_{\\mathbf{L}(R)} d\\mathbf{X},", + "$$", + "which we can evaluate as an iterated integral in which the first", + "integration is with respect to $x_i$. For example, if $i=1$, then", + "\\begin{equation}\\label{eq:7.3.19}", + "V(\\mathbf{L}(R))=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", + "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1+ax_j}_{a_1+ax_j} dx_1.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Since", + "$$", + "\\int^{b_1+ax_j}_{a_1+ax_j} dy_1=\\int^{b_1}_{a_1} dy_1,", + "$$", + "\\eqref{eq:7.3.19} can be rewritten as", + "\\begin{eqnarray*}", + "V(\\mathbf{L}(R))\\ar=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", + "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} dx_1\\\\", + "\\ar=(b_n-a_n)(b_{n-1}-a_{n-1})\\cdots (b_1-a_1)=V(R).", + "\\end{eqnarray*}", + " Hence,", + "$V(\\mathbf{L}(R))=|\\det(\\mathbf{E})|V(R)$,", + "since $\\det(\\mathbf{E})=1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "\\vskip12pt", + " \\centereps{3.6in}{4.6in}{fig070301.eps}", + " \\vskip6pt", + " \\refstepcounter{figure}", + " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.1}", + " \\vskip12pt", + "From what we have shown so far, \\eqref{eq:7.3.14} holds if $\\mathbf{A}$ is an", + "elementary matrix and $S$ is any compact Jordan measurable set. If", + "$\\mathbf{A}$ is an arbitrary nonsingular matrix,", + "\\newpage", + "\\noindent", + "\\hskip -.0em", + "then we can write $\\mathbf{A}$", + "as a product of elementary matrices \\eqref{eq:7.3.10} and apply our known", + "result successively to $\\mathbf{L}_1$, $\\mathbf{L}_2$, \\dots, $\\mathbf{L}_k$", + "(see", + "\\eqref{eq:7.3.13}). This yields", + "$$", + "V(\\mathbf{L}(S))=|\\det(\\mathbf{E}_k)|\\,|\\det(\\mathbf{E}_{k-1})|\\cdots", + "|\\det\\mathbf{E}_1| V(S)=|\\det(\\mathbf{A})|V(S),", + "$$", + "by Theorem~\\ref{thmtype:6.1.9} and induction." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" + ], + "ref_ids": [ + 215, + 216, + 173 + ] + } + ], + "ref_ids": [] + }, + { + "id": 217, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.8", + "categories": [], + "title": "", + "contents": [ + "\\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and", + "$f$ is continuous on $\\mathbf{G}(S).$ Then", + "\\begin{equation}\\label{eq:7.3.28}", + "\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_S f(\\mathbf{G}(\\mathbf{Y}))", + "|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $s$ be the edge length of $C$. Let $\\mathbf{Y}_0=", + "(c_1,c_2,\\dots,c_n)$ be the center of $C$, and suppose that", + " $\\mathbf{H}=(y_1,y_2,\\dots,y_n)\\in C$.", + "If $\\mathbf{H}= (h_1,h_2,\\dots,h_n)$ is continuously differentiable on", + "$C$, then applying the mean value theorem", + "(Theorem~\\ref{thmtype:5.4.5}) to the components of", + "$\\mathbf{H}$ yields", + "$$", + "h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)=\\sum_{j=1}^n", + "\\frac{\\partial h_i(\\mathbf{Y}_i)}{\\partial y_j}(y_j-c_j),\\quad 1\\le i\\le n,", + "$$", + "where $\\mathbf{Y}_i\\in C$. Hence, recalling that", + "$$", + "\\mathbf{H}'(\\mathbf{Y})=\\left[\\frac{\\partial h_i}{\\partial", + "y_j}\\right]_{i,j=1}^n,", + "$$", + "applying Definition~\\ref{thmtype:7.3.9}, and noting that $|y_j-c_j|\\le", + "s/2$, $1\\le j\\le n$, we infer that", + "$$", + "|h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)|\\le \\frac{s}{2}", + "\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C},\\quad 1\\le i\\le", + "n.", + "$$", + "This means that $\\mathbf{H}(C)$ is", + "contained in a cube with center $\\mathbf{X}_0=\\mathbf{H}(\\mathbf{Y}_0)$ and edge", + " length", + "$$", + "s\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}.", + "$$", + "Therefore,", + "\\begin{equation}\\label{eq:7.3.30}", + "\\begin{array}{rcl}", + "V(\\mathbf{H}(C))\\ar\\le", + "\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in", + "C} s^n\\\\[2\\jot]", + "\\ar=\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in C}", + "V(C).", + "\\end{array}", + "\\end{equation}", + "Now let", + "$$", + "\\mathbf{L}(\\mathbf{X})=\\mathbf{A}^{-1}\\mathbf{X}", + "$$", + "and set $\\mathbf{H}=\\mathbf{L}\\circ\\mathbf{G}$; then", + "$$", + "\\mathbf{H}(C)=\\mathbf{L}(\\mathbf{G}(C))", + "\\mbox{\\quad and\\quad}\\mathbf{H}'=\\mathbf{A}^{-1}\\mathbf{G}',", + "$$", + "so \\eqref{eq:7.3.30} implies that", + "\\begin{equation}\\label{eq:7.3.31}", + "V(\\mathbf{L}(\\mathbf{G}(C)))\\le", + "\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", + "\\right]^nV(C).", + "\\end{equation}", + "Since $\\mathbf{L}$ is linear,", + "Theorem~\\ref{thmtype:7.3.7} with $\\mathbf{A}$ replaced by $\\mathbf{A}^{-1}$ implies that", + "$$", + "V(\\mathbf{L}(\\mathbf{G}(C)))=|\\det(\\mathbf{A})^{-1}|V(\\mathbf{G}(C)).", + "$$", + "This and \\eqref{eq:7.3.31} imply that", + "$$", + "|\\det(\\mathbf{A}^{-1})|V(\\mathbf{G}(C))", + "\\le\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in", + "C}", + "\\right]^nV(C).", + "$$", + "Since $\\det(\\mathbf{A}^{-1})=1/\\det(\\mathbf{A})$, this", + "implies \\eqref{eq:7.3.29}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.7" + ], + "ref_ids": [ + 164, + 365, + 216 + ] + } + ], + "ref_ids": [] + }, + { + "id": 218, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.15", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}: \\E^n\\to \\R^n$ is continuously", + "differentiable on a bounded open set $N$ containing the compact", + "Jordan measurable set $S,$ and regular on $S^0.$ Suppose also that", + "$\\mathbf{G}(S)$ is Jordan measurable$,$", + "$f$ is continuous on $\\mathbf{G}(S),$ and $G(C)$ is Jordan measurable for", + "every cube $C\\subset N$. Then", + "\\begin{equation}\\label{eq:7.3.50}", + "\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $f$ is continuous on $\\mathbf{G}(S)$ and", + " $(|J\\mathbf{G}|) f\\circ\\mathbf{G}$ is continuous on $S$, the integrals", + "in \\eqref{eq:7.3.50} both exist, by", + "Corollary~\\ref{thmtype:7.3.2}.", + "Now let", + "$$", + "\\rho=\\dist\\ (\\partial S, N^c)", + "$$", + "(Exercise~5.1.25), and", + "$$", + "P=\\set{\\mathbf{Y}}{\\dist(\\mathbf{Y}, \\partial S)}\\le", + "\\frac{\\rho}{2}.", + "$$", + " Then $P$ is a", + "compact subset of $N$ (Exercise~5.1.26) and", + "$\\partial S\\subset P^0$", + "(Figure~\\ref{figure:7.3.4}).", + " Since $S$ is Jordan measurable, $V(\\partial S)=0$, by", + "Theorem~\\ref{thmtype:7.3.1}. Therefore,", + "if $\\epsilon>0$, we can choose cubes $C_1$, $C_2$, \\dots, $C_k$", + " in $P^0$ such that", + "\\begin{equation} \\label{eq:7.3.51}", + "\\partial S\\subset\\bigcup_{j=1}^k C_j^0", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.3.52}", + "\\sum_{j=1}^k V(C_j)<\\epsilon", + "\\end{equation}", + " Now let $S_1$ be the closure of the set of points in $S$", + "that are not in any of the cubes $C_1$, $C_2$, \\dots, $C_k$; thus,", + "$$", + "S_1=\\overline{S\\cap\\left(\\cup_{j=1}^k C_j\\right)^c}.", + "$$", + "\\newpage", + "\\noindent", + "Because of \\eqref{eq:7.3.51}, $S_1\\cap \\partial S=\\emptyset$,", + "so $S_1$ is a compact Jordan measurable subset of $S^0$. Therefore,", + "$\\mathbf{G}$ is regular on $S_1$, and $f$ is continuous on", + "$\\mathbf{G}(S_1)$.", + "Consequently, if $Q$ is as defined in \\eqref{eq:7.3.37}, then $Q(S_1)=0$", + "by Theorem~\\ref{thmtype:7.3.8}.", + " \\vskip12pt", + " \\centereps{2.1in}{2.8in}{fig070304.eps}", + " \\vskip6pt", + " \\refstepcounter{figure}", + " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.4}", + " \\vskip12pt", + "Now", + "\\begin{equation}\\label{eq:7.3.53}", + "Q(S)=Q(S_1)+Q(S\\cap S_1^c)=Q(S\\cap S_1^c)", + "\\end{equation}", + "(Exercise~\\ref{exer:7.3.11}) and", + "$$", + "|Q(S\\cap S_1^c)|\\le\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|+\\left|", + "\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y}))|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\right|.", + "$$", + " But", + "\\begin{equation} \\label{eq:7.3.54}", + "\\left|\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,", + "d\\mathbf{Y}\\right|\\le M_1M_2 V(S\\cap S_1^c),", + "\\end{equation}", + "where $M_1$ and $M_2$ are as defined in \\eqref{eq:7.3.38} and", + "\\eqref{eq:7.3.39}. Since", + "$S\\cap S_1^c\\subset \\cup_{j=1}^k C_j$,", + "\\eqref{eq:7.3.52} implies that $V(S\\cap S_1^k)<\\epsilon$; therefore,", + "\\begin{equation} \\label{eq:7.3.55}", + "\\left|\\int_{S\\cap S_1^c} f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,", + "d\\mathbf{Y}\\right|\\le M_1M_2\\epsilon,", + "\\end{equation}", + "from \\eqref{eq:7.3.54}. Also", + "\\begin{equation}\\label{eq:7.3.56}", + "\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le M_2", + "V(\\mathbf{G}(S\\cap S_1^c))\\le M_2\\sum_{j=1}^k V(\\mathbf{G}(C_j)).", + "\\end{equation}", + "\\newpage", + "\\noindent", + "By the argument that led to \\eqref{eq:7.3.30} with", + "${\\bf H}={\\bf G}$ and $C=C_{j}$,", + "$$", + "V(\\mathbf{G}(C_j))\\le\\left[\\max\\set{\\|\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}", + "{\\mathbf{Y}\\in C_j}\\right]^nV(C_j),", + "$$", + "so \\eqref{eq:7.3.56} can be rewritten as", + "$$", + "\\left|\\int_{\\mathbf{G}(S\\cap S_1^c)} f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le M_2", + "\\left[\\max\\set{\\|\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in P}", + "\\right]^n\\epsilon,", + "$$", + "because of \\eqref{eq:7.3.52}. Since $\\epsilon$ can be made arbitrarily", + "small, this and \\eqref{eq:7.3.55} imply that $Q(S\\cap S_1^c)=0$. Now", + "$Q(S)=0$, from \\eqref{eq:7.3.53}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.3.2", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.1", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.8" + ], + "ref_ids": [ + 297, + 214, + 217 + ] + } + ], + "ref_ids": [] + }, + { + "id": 219, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.4", + "categories": [], + "title": "", + "contents": [ + "If $(A,N)$ is a normed vector space$,$ then", + "\\begin{equation} \\label{eq:8.1.1}", + "\\rho(x,y)=N(x-y)", + "\\end{equation}", + "is a metric on $A.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From \\part{a} with $u=x-y$, $\\rho(x,y)=N(x-y)\\ge0$, with equality", + "if and only if $x=y$. From \\part{b} with $u=x-y$ and $a=-1$,", + "$$", + "\\rho(y,x)=N(y-x)=N(-(x-y))=N(x-y)=\\rho(x,y).", + "$$", + "From \\part{c} with $u=x-z$ and $v=z-y$,", + "$$", + "\\rho(x,y)=N(x-y)\\le N(x-z)+N(z-y)=\\rho(x,z)+\\rho(z,y).", + "$$", + "\\vskip-2em" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 220, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.5", + "categories": [], + "title": "", + "contents": [ + "If $x$ and $y$ are vectors in a normed vector space $(A,N),$ then", + "\\begin{equation} \\label{eq:8.1.2}", + "|N(x)-N(y)|\\le N(x-y).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since", + "$$", + "x=y+(x-y),", + "$$", + "Definition~\\ref{thmtype:8.1.3}\\part{c} with $u=y$ and $v=x-y$ implies that", + "$$", + "N(x)\\le N(y)+N(x-y),", + "$$", + "or", + "$$", + "N(x)-N(y)\\le N(x-y).", + "$$", + "Interchanging $x$ and $y$ yields", + "$$", + "N(y)-N(x)\\le N(y-x).", + "$$", + "Since $N(x-y)=N(y-x)$ (Definition~\\ref{thmtype:8.1.3}\\part{b} with", + "$u=x-y$ and $a=-1$), the last two inequalities imply \\eqref{eq:8.1.2}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:8.1.3" + ], + "ref_ids": [ + 368, + 368 + ] + } + ], + "ref_ids": [] + }, + { + "id": 221, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.9", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{X}\\in\\R^n$ and $p_2>p_1\\ge1,$ then", + "\\begin{equation} \\label{eq:8.1.12}", + "\\|\\mathbf{X}\\|_{p_2}\\le\\|\\mathbf{X}\\|_{p_1};", + "\\end{equation}", + "moreover,", + "\\begin{equation} \\label{eq:8.1.13}", + "\\lim_{p\\to\\infty}\\|\\mathbf{X}\\|_{p}=\\max\\set{|x_i|}{1\\le i\\le n}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $u_1$, $u_2$, \\dots, $u_n$ be", + "nonnegative and $M=\\max\\set{u_i}{1\\le i\\le n}$. Define", + "$$", + "\\sigma(p)=\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}.", + "$$", + "Since $u_i/\\sigma(p)\\le1$ and $p_2>p_1$,", + "$$", + "\\left(\\frac{u_i}{\\sigma(p_2)}\\right)^{p_1}\\ge", + "\\left(\\frac{u_i}{\\sigma(p_2)}\\right)^{p_2};", + "$$", + " therefore,", + "$$", + "\\frac{\\sigma(p_1)}{\\sigma(p_2)}", + "=\\left(\\sum_{i=1}^n\\left(\\frac{", + "u_i}{\\sigma(p_2)}\\right)^{p_1}\\right)^{1/p_1}", + "\\ge\\left(\\sum_{i=1}^n\\left(\\frac{", + "u_i}{\\sigma(p_2)}\\right)^{p_2}\\right)^{1/p_1}=1,", + "$$", + "so $\\sigma(p_1)\\ge\\sigma(p_2)$.", + "Since $M\\le\\sigma(p)\\le Mn^{1/p}$,", + "$\\lim_{p\\to\\infty}\\sigma(p)= M$.", + "Letting $u_i=|x_i|$ yields \\eqref{eq:8.1.12} and \\eqref{eq:8.1.13}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 222, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.11", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + " The union of open sets is open.", + "\\item % (b)", + " The intersection of closed sets is closed.", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 223, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.13", + "categories": [], + "title": "", + "contents": [ + "contains all its limit points$.$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 224, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.15", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + "The limit of a convergent sequence is unique$.$", + "\\item % (b)", + "If $\\lim_{n\\to\\infty}u_n=u,$ then every subsequence of", + "$\\{u_n\\}$ converges to $u.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 225, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.17", + "categories": [], + "title": "", + "contents": [ + "If a sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is convergent$,$", + "then it is a Cauchy sequence." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $\\lim_{n\\to\\infty}u_n=u$. If $\\epsilon>0$, there is an integer", + "$N$ such that", + "$\\rho(u_n,u)<\\epsilon/2$ if $n>N$. Therefore, if $m$, $n>N$, then", + "$$", + "\\rho(u_n,u_m)\\le\\rho(u_n,u)+\\rho(u,u_m)<\\epsilon.", + "$$", + "\\vskip-2em" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 226, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.19", + "categories": [], + "title": "The Principle of Nested Sets", + "contents": [ + "A metric space $(A,\\rho)$ is complete if and only if every", + "nested sequence", + "$\\{T_n\\}$ of nonempty closed subsets of $A$ such that", + " $\\lim_{n\\to\\infty}d(T_n)=0$", + "has a nonempty intersection$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $(A,\\rho)$ is complete and $\\{T_n\\}$", + "is a nested sequence", + " of nonempty closed subsets of $A$ such that", + " $\\lim_{n\\to\\infty}d(T_n)=0$.", + "For each $n$, choose", + " $t_n\\in T_n$. If $m\\ge n$,", + "then $t_m$, $t_n\\in T_n$, so $\\rho(t_n,t_m)1$ and we have specified $n_1$, $n_2$, \\dots, $n_{j-1}$", + "and", + "$T_1$, $T_2$, \\dots, $T_{j-1}$. Choose $n_j>n_{j-1}$ so that", + "$\\rho(t_n,t_{n_j})<2^{-j}$ if $n\\ge n_j$, and let", + "$T_j=\\set{t}{\\rho(t,t_{n_j})\\le2^{-j+1}}$. Then $T_j$ is closed", + "and nonempty, $T_{j+1}\\subset T_j$ for all $j$, and", + "$\\lim_{j\\to\\infty}d(T_j)=0$. Moreover, $t_n\\in T_j$ if $n\\ge n_j$.", + "Therefore, if $\\overline t\\in\\cap_{j=1}^\\infty T_j$, then", + "$\\rho(t_n,\\overline t)<2^{-j}$, $n\\ge n_j$, so", + "$\\lim_{n\\to\\infty}t_n=\\overline t$, contrary to our assumption.", + "Hence, $\\cap_{j=1}^\\infty T_j=\\emptyset$.", + "\\boxit{Equivalent Metrics}", + "When considering more than one metric on a given set $A$", + "we must be careful, for example, in saying that a set is open,", + "or that a sequence converges, etc., since the truth or falsity", + "of the statement will in general depend on the metric as well as the", + "set on which it is imposed. In this situation we will alway", + "refer to the metric space by its ``full name;\" that is, $(A,\\rho)$", + "rather than just $A$.", + "\\begin{definition} \\label{thmtype:8.1.20}", + "If $\\rho$ and $\\sigma$ are both metrics on a set $A$, then $\\rho$", + "and $\\sigma$ are {\\it equivalent \\/}", + "\\hskip-.2em if there are positive constants $\\alpha$ and $\\beta$", + "such that", + "\\begin{equation} \\label{eq:8.1.18}", + "\\alpha\\le\\frac{\\rho(x,y)}{\\sigma(x,y)}\\le\\beta", + "\\mbox{\\quad for all \\quad}x,y\\in A\\mbox{\\quad such that \\quad}x\\ne y.", + "\\end{equation}", + "\\end{definition}" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 227, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.21", + "categories": [], + "title": "", + "contents": [ + "If $\\rho$ and $\\sigma$ are equivalent metrics on a set $A,$ then", + " $(A,\\rho)$ and $(A,\\sigma)$ have the same open sets." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that \\eqref{eq:8.1.18} holds. Let $S$ be an open set in", + "$(A,\\rho)$ and let $x_0\\in S$. Then there is an $\\epsilon>0$ such", + "that $x\\in S$ if $\\rho(x,x_0)<\\epsilon$, so the second", + "inequality in \\eqref{eq:8.1.18}", + "implies that $x_0\\in S$ if $\\sigma(x,x_0)\\le\\epsilon/\\beta$.", + "Therefore, $S$ is open in $(A,\\sigma)$.", + "Conversely, suppose that $S$ is open in $(A,\\sigma)$", + "and let $x_0\\in S$. Then there is an $\\epsilon>0$ such", + "that $x\\in S$ if $\\sigma(x,x_0)<\\epsilon$, so the first", + "inequality in \\eqref{eq:8.1.18}", + "implies that $x_0\\in S$ if $\\rho(x,x_0)\\le\\epsilon\\alpha$.", + "Therefore, $S$ is open in $(A,\\rho)$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 228, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.22", + "categories": [], + "title": "", + "contents": [ + "Any two norms $N_1$ and $N_2$ on $\\R^n$ induce equivalent", + "metrics on~$\\R^n.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "It suffices to show that there are positive constants $\\alpha$", + "and $\\beta$ such", + "\\begin{equation} \\label{eq:8.1.19}", + "\\alpha\\le\\frac{N_1(\\mathbf{X})}{N_2{(\\bf X})}\\le\\beta\\mbox{\\quad if", + "\\quad}", + "\\mathbf{X}\\ne\\mathbf{0}.", + "\\end{equation}", + "We will show that if $N$ is any norm on $\\R^n$, there are", + "positive constants $a_N$ and $b_N$ such that", + "\\begin{equation} \\label{eq:8.1.20}", + "a_N\\|\\mathbf{X}\\|_2\\le N(\\mathbf{X})\\le b_N\\|\\mathbf{X}\\|_2 \\mbox{\\quad if", + "\\quad}", + "\\mathbf{X}\\ne\\mathbf{0}", + "\\end{equation}", + "and leave it to you to verify that this implies \\eqref{eq:8.1.19}", + "with $\\alpha=a_{N_1}/b_{N_2}$ and $\\beta=b_{N_1}/a_{N_2}$.", + "We write $\\mathbf{X}-\\mathbf{Y}=(x_1,x_2, \\dots,x_n)$ as", + "$$", + "\\mathbf{X}-\\mathbf{Y}=\\sum_{i=1}^n\\,(x_i-y_i)\\mathbf{E}_i,", + "$$", + "where $\\mathbf{E}_i$ is the vector with $i$th component equal to $1$", + "and all other components equal to $0$. From", + "Definition~\\ref{thmtype:8.1.3}\\part{b}, \\part{c}, and induction,", + "$$", + "N(\\mathbf{X}-\\mathbf{Y})\\le\\sum_{i=1}^n|x_i-y_i|N(\\mathbf{E_i});", + "$$", + "therefore, by Schwarz's inequality,", + "\\begin{equation} \\label{eq:8.1.21}", + "N(\\mathbf{X}-\\mathbf{Y})\\le K\\|\\mathbf{X}-\\mathbf{Y}\\|_2,", + "\\end{equation}", + "where", + "$$", + "K=\\left(\\sum_{i=1}^nN^2(\\mathbf{E_i})\\right)^{1/2}.", + "$$", + "From \\eqref{eq:8.1.21} and Theorem~\\ref{thmtype:8.1.5},", + "$$", + "|N(\\mathbf{X})-N(\\mathbf{Y})|\\le K\\|\\mathbf{X}-\\mathbf{Y}\\|_2,", + "$$", + "so $N$ is continuous on $\\R_2^n=\\R^n$.", + "By Theorem~\\ref{thmtype:5.2.12}, there are vectors", + "$\\mathbf{U}_1$ and $\\mathbf{U}_2$ such that $\\|\\mathbf{U}_1\\|_2=", + "\\|\\mathbf{U}_2\\|_2=1$,", + "$$", + "N(\\mathbf{U}_1)=\\min\\set{N(\\mathbf{U})}{\\|\\mathbf{U}\\|_2=1},", + "\\mbox{\\quad and \\quad}", + "N(\\mathbf{U}_2)=\\max\\set{N(\\mathbf{U})}{\\|\\mathbf{U}\\|_2=1}.", + "$$", + "If", + "$a_N=N(\\mathbf{U}_1)$ and $b_N=N(\\mathbf{U}_2)$, then", + "$a_N$ and $b_N$ are positive", + "(Definition~\\ref{thmtype:8.1.3}\\part{a}), and", + "$$", + "a_N\\le N\\left(\\frac{\\mathbf{X}}{\\|\\mathbf{X}\\|_2}\\right)\\le b_N", + "\\mbox{\\quad if \\quad} \\mathbf{X}\\ne\\mathbf{0}.", + "$$", + "This and Definition~\\ref{thmtype:8.1.3}\\part{b} imply", + "\\eqref{eq:8.1.20}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:8.1.5", + "TRENCH_REAL_ANALYSIS-thmtype:5.2.12", + "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:8.1.3" + ], + "ref_ids": [ + 368, + 220, + 152, + 368, + 368 + ] + } + ], + "ref_ids": [] + }, + { + "id": 229, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.23", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\rho$ and $\\sigma$ are equivalent metrics on $A.$ Then", + "\\begin{alist}", + "\\item % (a)", + "A sequence $\\{u_n\\}$ converges to $u$ in $(A,\\rho)$ if and only", + "if it converges to $u$ in~$(A,\\sigma).$", + "\\item % (a)", + "A sequence $\\{u_n\\}$ is a Cauchy sequence in $(A,\\rho)$ if and only", + "if it is a Cauchy sequence in $(A,\\sigma).$", + "\\item % (b)", + "$(A,\\rho)$ is complete if and only if $(A,\\sigma)$ is complete$.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 230, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.3", + "categories": [], + "title": "", + "contents": [ + "An infinite subset $T$ of $A$ is compact", + "if and only if every infinite subset of $T$ has a limit point in $T.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $T$ has an infinite", + "subset $E$ with no limit point in $T$. Then, if $t\\in T$,", + " there is an open set $H_t$ such that $t\\in H_t$ and $H_t$", + "contains at most one member of $E$. Then ${\\mathcal", + "H}=\\cup\\set{H_t}{t\\in T}$ is an open covering of $T$, but", + " no finite collection $\\{H_{t_1},H_{t_2}, \\dots,H_{t_k}\\}$ of sets", + "from ${\\mathcal H}$ can cover $E$, since $E$ is infinite. Therefore, no", + "such collection can cover $T$;", + "that is, $T$ is not compact.", + "Now suppose that every infinite subset of $T$ has a limit point in", + "$T$, and let", + "${\\mathcal H}$ be an open covering of $T$.", + "We first show that there is a sequence", + "$\\{H_i\\}_{i=1}^\\infty$ of sets from ${\\mathcal H}$ that covers $T$.", + "If $\\epsilon>0$, then $T$ can be covered by", + " $\\epsilon$-neighborhoods of finitely many points of $T$.", + "We prove this by contradiction.", + "Let $t_1\\in T$. If", + "$N_\\epsilon(t_1)$ does not cover $T$, there is a $t_2\\in T$ such", + "that", + "$\\rho(t_1,t_2)\\ge\\epsilon$.", + "Now suppose that $n\\ge 2$ and we have chosen $t_1$, $t_2$, \\dots, $t_n$", + "such that $\\rho(t_i,t_j)\\ge\\epsilon$, $1\\le i0$ such that", + "$N_\\epsilon(t)\\subset H$. Since $t\\in G_j$ for infinitely", + "many values of $j$ and $\\lim_{j\\to\\infty}d(G_j)=0$,", + "$$", + "G_j\\subset N_\\epsilon(t)\\subset H", + "$$", + "for some $j$. Therefore,", + "if $\\{G_{j_i}\\}_{i=1}^\\infty$", + "is the subsequence of $\\{G_j\\}$ such that $G_{j_i}$ is a subset of", + "some $H_i$ in ${\\mathcal H}$ (the $\\{H_i\\}$ are not", + "necessarily distinct), then", + "\\begin{equation} \\label{eq:8.2.1}", + "T\\subset\\bigcup_{i=1}^\\infty H_i.", + "\\end{equation}", + "We will now show that", + "\\begin{equation} \\label{eq:8.2.2}", + "T\\subset \\bigcup_{i=1}^N H_i.", + "\\end{equation}", + "for some integer $N$. If this is not so, there is an infinite", + "sequence $\\{t_n\\}_{n=1}^\\infty$ in $T$ such that", + "\\begin{equation} \\label{eq:8.2.3}", + "t_n\\notin \\bigcup_{i=1}^n H_i, \\quad n\\ge 1.", + "\\end{equation}", + "From our assumption,", + " $\\{t_n\\}_{n=1}^\\infty$", + "has a limit $\\overline t$ in $T$. From \\eqref{eq:8.2.1},", + "$\\overline t\\in H_k$ for some $k$, so", + "$N_\\epsilon(\\overline t)\\subset H_k$ for some $\\epsilon>0$. Since", + "$\\lim_{n\\to\\infty}t_n=\\overline t$, there is an integer $N$ such that", + "$$", + "t_n\\in N_\\epsilon(\\overline t)\\subset H_k\\subset \\bigcup_{i=1}^nH_i,\\quad", + "n>k,", + "$$", + "which contradicts \\eqref{eq:8.2.3}. This verifies \\eqref{eq:8.2.2},", + "so $T$ is compact." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 231, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", + "categories": [], + "title": "", + "contents": [ + "A subset $T$ of a metric $A$ is compact if and only if", + "every infinite sequence $\\{t_n\\}$ of members of $T$ has a", + "subsequence that converges to a member of $T.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $T$ is compact and $\\{t_n\\}\\subset T$. If $\\{t_n\\}$", + "has only finitely many distinct terms, there is a $\\overline t$", + "in $T$ such that $t_n=\\overline t$ for infinitely many values of $n$;", + "if this is so for $n_10$. Since $\\{t_n\\}$ is a Cauchy sequence,", + "there is an integer $N$ such that $\\rho(t_n,t_m)<\\epsilon$,", + " $n>m\\ge N$. From \\eqref{eq:8.2.4},", + "there is an $m=n_j\\ge N$ such that $\\rho(t_m,\\overline t)<\\epsilon$.", + "Therefore,", + "$$", + "\\rho(t_n,\\overline t)\\le \\rho(t_n,t_m)+\\rho(t_m,\\overline", + "t)<2\\epsilon,\\quad n\\ge m.", + "$$", + "\\vskip-2em" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" + ], + "ref_ids": [ + 231 + ] + } + ], + "ref_ids": [] + }, + { + "id": 233, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.6", + "categories": [], + "title": "", + "contents": [ + "If $T$ is", + "compact$,$ then $T$ is closed and bounded." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that", + " $\\overline t$ is a limit point of $T$. For each $n$, choose", + "$t_n\\ne\\overline t\\in", + "B_{1/n}(\\overline t)\\cap T$. Then $\\lim_{n\\to\\infty}t_n=\\overline t$.", + "Since every subsequence of $\\{t_n\\}$ also converges to $\\overline t$,", + " $\\overline t\\in T$, by", + "Theorem~\\ref{thmtype:8.2.3}. Therefore, $T$ is closed.", + "The family of unit open balls", + "${\\mathcal H}=\\set{B_1(t)}{t\\in T}$", + "is an open covering of $T$. Since $T$ is compact, there are", + "finitely many members $t_1$, $t_2$, \\dots, $t_n$ of $T$ such that", + "$S\\subset \\cup_{j=1}^nB_1(t_j)$. If $u$ and $v$ are arbitrary", + "members of $T$, then $u\\in B_1(t_r)$ and $v\\in B_1(t_s)$ for some", + "$r$ and $s$ in $\\{1,2, \\dots,n\\}$, so", + "\\begin{eqnarray*}", + "\\rho(u,v)\\ar\\le \\rho(u,t_r)+\\rho(t_r,t_s)+\\rho(t_s,v)\\\\", + "\\ar\\le 2+\\rho(t_r,t_s)\\le2+\\max\\set{\\rho(t_i,t_j)}{1\\le i0$", + "such that there is no finite $\\epsilon$-net for $T$.", + "Let $t_1\\in T$. Then there must be a $t_2$ in $T$", + "such that $\\rho(t_1,t_2)>\\epsilon$. (If not, the singleton", + "set $\\{t_1\\}$ would be a finite $\\epsilon$-net for $T$.)", + "Now suppose that $n\\ge 2$ and we have chosen $t_1$, $t_2$, \\dots, $t_n$", + "such that $\\rho(t_i,t_j)\\ge\\epsilon$, $1\\le i1$ and we have chosen", + "an infinite subsequence $\\{s_{i,n-1}\\}_{i=1}^\\infty$ of", + "$\\{s_{i,n-2}\\}_{i=1}^\\infty$.", + "Since $T_{1/n}$ is finite and $\\{s_{i,n-1}\\}_{i=1}^\\infty$", + "is infinite,", + "there must be member $t_n$ of $T_{1/n}$ such that", + "$\\rho(s_{i,n-1},t_n)\\le1/n$ for infinitely many values of $i$.", + "Let $\\{s_{in}\\}_{i=1}^\\infty$ be the subsequence of", + "$\\{s_{i,n-1}\\}_{i=1}^\\infty$ such that $\\rho(s_{in},t_n)\\le1/n$.", + "From the triangle inequality,", + "\\begin{equation} \\label{eq:8.2.5}", + "\\rho(s_{in},s_{jn})\\le2/n,\\quad i,j\\ge1,\\quad n\\ge 1.", + "\\end{equation}", + "Now let $\\widehat s_i=s_{ii}$, $i\\ge 1$. Then $\\{\\widehat s_i\\}_{i=1}^\\infty$", + "is an infinite sequence of members of $T$. Moroever, if", + "$i,j\\ge n$, then $\\widehat s_i$ and $\\widehat s_j$ are both included in", + "$\\{s_{in}\\}_{i=1}^\\infty$, so \\eqref{eq:8.2.5} implies that", + "$\\rho(\\widehat s_i,\\widehat s_j)\\le2/n$; that is, $\\{\\widehat s_i\\}_{i=1}^\\infty$", + "is a Cauchy sequence and therefore has a limit, since $(A,\\rho)$", + " is complete." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" + ], + "ref_ids": [ + 231 + ] + } + ], + "ref_ids": [] + }, + { + "id": 236, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.11", + "categories": [], + "title": "", + "contents": [ + "A nonempty subset $T$ of $C[a,b]$ is compact if and only if", + "it is closed$,$ uniformly bounded$,$ and equicontinuous." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For necessity, suppose that $T$ is compact. Then $T$ is closed", + "(Theorem~\\ref{thmtype:8.2.6}) and totally bounded", + "(Theorem~\\ref{thmtype:8.2.8}). Therefore, if $\\epsilon>0$, there is", + "a finite subset $T_\\epsilon=\\{g_1,g_2, \\dots,g_k\\}$ of $C[a,b]$", + "such that if $f\\in T$, then", + "$\\|f-g_i\\|\\le \\epsilon$", + "for some $i$ in $\\{1,2, \\dots,k\\}$.", + "If we temporarily let $\\epsilon=1$, this implies that", + "$$", + "\\|f\\|=\\|(f-g_i)+g_i\\|\\le\\|f-g_i\\|+\\|g_i\\|\\le 1+\\|g_i\\|,", + "$$", + "which implies \\eqref{eq:8.2.6} with", + "$$", + "M=1+\\max\\set{\\|g_i\\|}{1\\le i\\le k}.", + "$$", + "For \\eqref{eq:8.2.7}, we again let $\\epsilon$ be arbitary, and write", + "\\begin{equation} \\label{eq:8.2.8}", + "\\begin{array}{rcl}", + "|f(x_1)-f(x_2)|", + "\\ar\\le |f(x_1)-g_i(x_1)|+|g_i(x_1)-g_i(x_2)|+|g_i(x_2)-f(x_2)|\\\\", + "\\ar\\le |g_i(x_1)-g_i(x_2)|+2\\|f-g_i\\|\\\\", + "\\ar< |g_i(x_1)-g_i(x_2)|+2\\epsilon.", + "\\end{array}", + "\\end{equation}", + "Since each of the finitely many functions $g_1$, $g_2$, \\dots, $g_k$", + "is uniformly continuous on $[a,b]$", + "(Theorem~\\ref{thmtype:2.2.12}), there is a $\\delta>0$ such that", + "$$", + "|g_i(x_1)-g_i(x_2)|<\\epsilon\\mbox{\\quad if \\quad}", + "|x_1-x_2|<\\delta,\\quad 1\\le i\\le k.", + "$$", + "This and \\eqref{eq:8.2.8} imply \\eqref{eq:8.2.7} with $\\epsilon$", + "replaced by $3\\epsilon$. Since this replacement is of no consequence,", + "this proves necessity.", + "For sufficiency, we will show that $T$ is totally bounded.", + " Since $T$ is closed by assumption and", + "$C[a,b]$ is complete, Theorem~\\ref{thmtype:8.2.9} will then imply that", + "$T$ is compact.", + "Let $m$ and $n$ be positive integers and let", + "$$", + "\\xi_r=a+\\frac{r}{m}(b-a),\\quad 0\\le r\\le m,", + "\\mbox{\\quad and \\quad}", + "\\eta_s=\\frac{sM}{n},\\quad -n\\le s\\le n;", + "$$", + "that is, $a=\\xi_0<\\xi_1<\\cdots<\\xi_m=b$ is a partition of $[a,b]$", + "into subintervals of length $(b-a)/m$, and", + "$-M=\\eta_{-n}<\\eta_{-n+1}<\\cdots<\\eta_{n-1}<\\eta_n=M$ is a partition", + "of the \\phantom{segment}", + "\\newpage", + "\\noindent", + " segment of the $y$-axis", + "between $y=-M$ and $y=M$ into", + "subsegments of length $M/n$.", + "Let $S_{mn}$ be the subset of $C[a,b]$ consisting of functions $g$", + "such that", + "$$", + "\\{g(\\xi_0), g(\\xi_1), \\dots, g(\\xi_m)\\}", + "\\subset\\{\\eta_{-n},\\eta_{-n+1} \\dots,\\eta_{n-1}, \\eta_n\\}", + "$$", + " and $g$ is linear on", + " $[\\xi_{i-1},\\xi_i]$,", + "$1\\le i\\le m$.", + " Since there are only $(m+1)(2n+1)$", + "points", + "of the form $(\\xi_r,\\eta_s)$, $S_{mn}$ is a finite subset of", + "$C[a,b]$.", + "Now suppose that $\\epsilon>0$, and choose $\\delta>0$ to satisfy", + "\\eqref{eq:8.2.7}. Choose $m$ and $n$ so that $(b-a)/m<\\delta$", + "and $2M/n<\\epsilon$. If $f$ is an arbitrary member of $T$,", + "there is a $g$ in $S_{mn}$ such that", + "\\begin{equation} \\label{eq:8.2.9}", + "|g(\\xi_i)-f(\\xi_i)|<\\epsilon,\\quad", + "0\\le i\\le m.", + "\\end{equation}", + "If $0\\le i\\le m-1$,", + "\\begin{equation} \\label{eq:8.2.10}", + "|g(\\xi_i)-g(\\xi_{i+1})|=|g(\\xi_i)-f(\\xi_i)|+|f(\\xi_i)-f(\\xi_{i+1})|", + "+|f(\\xi_{i+1})-g(\\xi_{i+1})|.", + "\\end{equation}", + "Since $\\xi_{i+1}-\\xi_i<\\delta$, \\eqref{eq:8.2.7}, \\eqref{eq:8.2.9},", + "and \\eqref{eq:8.2.10} imply that", + "$$", + "|g(\\xi_i)-g(\\xi_{i+1})|<3\\epsilon.", + "$$", + "Therefore,", + "\\begin{equation} \\label{eq:8.2.11}", + "|g(\\xi_i)-g(x)|<3\\epsilon,\\quad \\xi_i\\le x\\le \\xi_{i+1},", + "\\end{equation}", + "since $g$ is linear on $[\\xi_i,\\xi_{i+1}]$.", + "Now let $x$ be an arbitrary point in $[a,b]$, and choose $i$", + "so that $x\\in[\\xi_i,\\xi_{i+1}]$. Then", + "$$", + "|f(x)-g(x)|\\le|f(x)-f(\\xi_i)|+|f(\\xi_i)-g(\\xi_i)|+|g(\\xi_i)-g(x)|,", + "$$", + "so \\eqref{eq:8.2.7}, \\eqref{eq:8.2.9}, and \\eqref{eq:8.2.11} imply that", + "$|f(x)-g(x)|<5\\epsilon$, $a\\le x\\le b$. Therefore,", + "$S_{mn}$ is a finite $5\\epsilon$-net for $T$, so $T$ is totally", + "bounded." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.6", + "TRENCH_REAL_ANALYSIS-thmtype:8.2.8", + "TRENCH_REAL_ANALYSIS-thmtype:2.2.12", + "TRENCH_REAL_ANALYSIS-thmtype:8.2.9" + ], + "ref_ids": [ + 233, + 234, + 25, + 235 + ] + } + ], + "ref_ids": [] + }, + { + "id": 237, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.12", + "categories": [], + "title": "", + "contents": [ + "Suppose that ${\\mathcal F}$ is an infinite uniformly bounded and equicontinuous", + "family of functions on $[a,b].$ Then there is a sequence $\\{f_n\\}$", + "in ${\\mathcal F}$ that converges uniformly to a continuous function", + " on $[a,b].$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $T$ be the closure of ${\\mathcal F}$; that is, $f\\in T$", + "if and only if either $f\\in T$ or $f$ is the uniform limit", + "of a sequence of members of ${\\mathcal F}$. Then $T$ is also", + "uniformly bounded and equicontinuous (verify),", + "and $T$ is closed. Hence, $T$ is compact, by", + "Theorem~\\ref{thmtype:8.2.12}. Therefore, ${\\mathcal F}$ has a limit point", + "in $T$. (In this context, the limit point is a function $f$ in", + "$T$.) Since $f$ is a limit point of ${\\mathcal F}$, there is for each", + "integer $n$ a function $f_n$ in ${\\mathcal F}$ such that $\\|f_n-f\\|<1/n$;", + "that is $\\{f_n\\}$ converges uniformly to $f$ on $[a,b]$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.12" + ], + "ref_ids": [ + 237 + ] + } + ], + "ref_ids": [] + }, + { + "id": 238, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.3", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\widehat u\\in\\overline D_f.$ Then", + "\\begin{equation} \\label{eq:8.3.3}", + "\\lim_{u\\to \\widehat u}f(u)=\\widehat v", + "\\end{equation}", + "if and only if", + "\\begin{equation} \\label{eq:8.3.4}", + "\\lim_{n\\to\\infty}f(u_n)=\\widehat v", + "\\end{equation}", + "for every sequence $\\{u_n\\}$ in $D_f$ such that", + "\\begin{equation} \\label{eq:8.3.5}", + "\\lim_{n\\to\\infty}u_n=\\widehat u.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that \\eqref{eq:8.3.3} is true, and let $\\{u_n\\}$ be a sequence in", + "$D_f$ that satisfies \\eqref{eq:8.3.5}. Let $\\epsilon>0$ and choose", + "$\\delta>0$ to satisfy \\eqref{eq:8.3.1}. From \\eqref{eq:8.3.5}, there is", + "an integer $N$ such that $\\rho(u_n,\\widehat u)<\\delta$ if $n\\ge N$.", + "Therefore, $\\sigma(f(u_n),\\widehat v)<\\epsilon$ if $n\\ge N$, which implies", + "\\eqref{eq:8.3.4}.", + "For the converse, suppose that \\eqref{eq:8.3.3} is false.", + "Then there is an $\\epsilon_0>0$ and a sequence $\\{u_n\\}$", + "in $D_f$ such that $\\rho(u_n,\\widehat u)<1/n$ and $\\sigma(f(u_n),\\widehat", + "v)\\ge\\epsilon_0$, so \\eqref{eq:8.3.4} is false.", + "\\mbox{}\\hfill" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 239, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.4", + "categories": [], + "title": "", + "contents": [ + "A function $f$ is continuous at $\\widehat u$ if and", + "only if", + "$$", + "\\lim_{u\\to\\widehat u} f(u)=f(\\widehat u).", + "$$" + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 240, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.5", + "categories": [], + "title": "", + "contents": [ + "A function $f$ is continuous at $\\widehat u$ if and", + "only if", + "$$", + "\\lim_{n\\to\\infty} f(u_n)=f(\\widehat u)", + "$$", + "whenever $\\{u_n\\}$ is a sequence in $D_f$ that converges to $\\widehat", + "u$." + ], + "refs": [], + "proofs": [], + "ref_ids": [] + }, + { + "id": 241, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.6", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a compact set $T,$ then $f(T)$ is compact." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $\\{v_n\\}$ be an infinite sequence in $f(T)$.", + "For each $n$, $v_n=f(u_n)$ for some $u_n\\in T$. Since $T$", + "is compact, $\\{u_n\\}$ has a subsequence", + "$\\{u_{n_j}\\}$ such that $\\lim_{j\\to\\infty}u_{n_j}=\\widehat u\\in T$", + "(Theorem~\\ref{thmtype:8.2.4}).", + "From Theorem~\\ref{thmtype:8.3.5},", + "$\\lim_{j\\to\\infty}f(u_{n_j})=f(\\widehat", + "u)$; that is, $\\lim_{j\\to\\infty}v_{n_j}=f(\\widehat u)$. Therefore, $f(T)$", + "is compact, again by", + "Theorem~\\ref{thmtype:8.2.4}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", + "TRENCH_REAL_ANALYSIS-thmtype:8.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:8.2.4" + ], + "ref_ids": [ + 231, + 240, + 231 + ] + } + ], + "ref_ids": [] + }, + { + "id": 242, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.8", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a compact set $T,$", + "then $f$ is uniformly continuous on $T$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $f$ is not uniformly continuous on $T$,", + "then for some", + "$\\epsilon_0>0$", + "there are sequences $\\{u_n\\}$ and $\\{v_n\\}$ in $T$ such that", + "$\\rho(u_n,v_n)<1/n$ and", + "\\begin{equation} \\label{eq:8.3.6}", + "\\sigma(f(u_n),f(v_n))\\ge\\epsilon_0.", + "\\end{equation}", + "Since $T$ is compact,", + " $\\{u_n\\}$ has a subsequence", + "$\\{u_{n_k}\\}$ that converges to a limit $\\widehat u$ in", + "$T$ (Theorem~\\ref{thmtype:8.2.4}). Since", + "$\\rho(u_{n_k},v_{n_k})<1/n_k$,", + "$\\lim_{k\\to\\infty}v_{n_k}=\\widehat u$ also.", + " Then", + "$$", + "\\lim_{k\\to\\infty}f(u_{n_k})=\\dst\\lim_{k\\to", + "\\infty}f(v_{n_k})=f(\\widehat u)", + "$$", + " (Theorem~~\\ref{thmtype:8.3.5}), which", + "contradicts \\eqref{eq:8.3.6}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.2.4", + "TRENCH_REAL_ANALYSIS-thmtype:8.3.5" + ], + "ref_ids": [ + 231, + 240 + ] + } + ], + "ref_ids": [] + }, + { + "id": 243, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.10", + "categories": [], + "title": "Contraction Mapping Theorem", + "contents": [ + "If $f$ is a contraction of a complete metric space $(A,\\rho),$", + "then the equation", + "\\begin{equation} \\label{eq:8.3.8}", + "f(u)=u", + "\\end{equation}", + "has a unique solution$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "To see that \\eqref{eq:8.3.8} cannot have more than one solution,", + "suppose that $u=f(u)$ and $v=f(v)$. Then", + "\\begin{equation} \\label{eq:8.3.9}", + "\\rho(u,v)=\\rho(f(u),f(v)).", + "\\end{equation}", + "However, \\eqref{eq:8.3.7} implies that", + "\\begin{equation} \\label{eq:8.3.10}", + "\\rho(f(u),f(v))\\le\\alpha\\rho(u,v).", + "\\end{equation}", + "Since \\eqref{eq:8.3.9} and \\eqref{eq:8.3.10} imply that", + "$$", + "\\rho(u,v)\\le\\alpha\\rho(u,v)", + "$$", + "and $\\alpha<1$, it follows that $\\rho(u,v)=0$. Hence $u=v$.", + "We will now show that \\eqref{eq:8.3.8} has a solution.", + "With $u_0$ arbitrary, define", + "\\begin{equation}\\label{eq:8.3.11}", + "u_n=f(u_{n-1}),\\quad n\\ge1.", + "\\end{equation}", + "We will show that $\\{u_n\\}$ converges. From \\eqref{eq:8.3.7} and", + "\\eqref{eq:8.3.11},", + "\\begin{equation} \\label{eq:8.3.12}", + "\\rho(u_{n+1},u_n)=\\rho(f(u_n),f(u_{n-1}))\\le\\alpha\\rho(u_n,u_{n-1}).", + "\\end{equation}", + "\\newpage", + "\\noindent", + "The inequality", + "\\begin{equation}\\label{eq:8.3.13}", + "\\rho(u_{n+1},u_n)\\le \\alpha^n \\rho(u_1,u_0),\\quad n\\ge0,", + "\\end{equation}", + "follows by induction from \\eqref{eq:8.3.12}. If $n>m$, repeated", + "application of the triangle inequality yields", + "$$", + "\\rho(u_n,u_m)", + "\\le", + "\\rho(u_n,u_{n-1})+\\rho(u_{n-1},u_{n-2})+\\cdots+\\rho(u_{m+1},u_m),", + "$$", + "and \\eqref{eq:8.3.13} yields", + "$$", + "\\rho(u_n,u_m)\\le\\rho(u_1,u_0)\\alpha^m(1+\\alpha+\\cdots+\\alpha^{n-m-1})<", + "\\frac{\\alpha^m}{1-\\alpha}.", + "$$", + "Now it follows that", + "$$", + "\\rho(u_n,u_m)<\\frac{\\rho(u_1,u_0)}{1-\\alpha}\\alpha^N\\mbox{\\quad", + "if\\quad} n,m>N,", + "$$", + "and, since $\\lim_{N\\to\\infty} \\alpha^N=0$, $\\{u_n\\}$ is a Cauchy", + "sequence. Since $A$ is complete, $\\{u_n\\}$ has a limit $\\widehat", + "u$. Since $f$ is continuous at", + "$\\widehat u$,", + "$$", + "f(\\widehat u)=\\lim_{n\\to\\infty}f(u_{n-1})=\\lim_{n\\to\\infty}u_n=\\widehat u,", + "$$", + "where Theorem~~\\ref{thmtype:8.3.5} implies the first equality and", + "\\eqref{eq:8.3.11} implies the second." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:8.3.5" + ], + "ref_ids": [ + 240 + ] + } + ], + "ref_ids": [] + }, + { + "id": 244, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.2", + "categories": [], + "title": "", + "contents": [ + "If $f$ is differentiable at $x_0,$ then", + "\\begin{equation}\\label{eq:2.3.3}", + "f(x)=f(x_0)+[f'(x_0)+E(x)](x-x_0),", + "\\end{equation}", + "where $E$ is defined on a neighborhood of $x_0$ and", + "$$", + "\\lim_{x\\to x_0} E(x)=E(x_0)=0.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Define", + "\\begin{equation} \\label{eq:2.3.4}", + "E(x)=\\left\\{\\casespace\\begin{array}{ll}", + "\\dst\\frac{f(x)-f(x_0)}{ x-x_0}-", + "f'(x_0),&x\\in D_f\\mbox{ and }x\\ne x_0,\\\\[2\\jot]", + "0,&x=x_0.", + "\\end{array}\\right.", + "\\end{equation}", + "Solving \\eqref{eq:2.3.4} for $f(x)$ yields \\eqref{eq:2.3.3} if $x\\ne x_0$,", + "and \\eqref{eq:2.3.3} is obvious if $x=x_0$.", + "Definition~\\ref{thmtype:2.3.1}", + "implies that $\\lim_{x\\to x_0}E(x)=0$. We defined $E(x_0)=0$ to make", + "$E$ continuous at $x_0$.", + "\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.3.1" + ], + "ref_ids": [ + 313 + ] + } + ], + "ref_ids": [] + }, + { + "id": 245, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.5.2", + "categories": [], + "title": "", + "contents": [ + "If $f^{(n)}(x_0)$ exists$,$ then", + "\\begin{equation}\\label{eq:2.5.7}", + "f(x)=\\sum_{r=0}^n\\frac{f^{(r)}(x_0)}{ r!} (x-x_0)^r+E_n(x)(x-x_0)^n,", + "\\end{equation}", + "where", + "$$", + "\\lim_{x\\to x_0} E_n(x)=E_n(x_0)=0.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Define", + "$$", + "E_n(x)=", + "\\left\\{\\casespace\\begin{array}{ll}", + "\\dst\\frac{f(x)-T_n(x)}{(x-x_0)^n},&x\\in D_f-\\{x_0\\},\\\\", + "0,&x=x_0.\\end{array}\\right.", + "$$", + "Then \\eqref{eq:2.5.5} implies that $\\lim_{x\\to x_0}E_n(x)=E_n(x_0)=0$,", + "and it is straightforward to verify \\eqref{eq:2.5.7}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 246, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", + "categories": [], + "title": "", + "contents": [ + "Suppose that", + "\\begin{equation} \\label{eq:3.2.1}", + "|f(x)|\\le M,\\quad a\\le x\\le b,", + "\\end{equation}", + "and let $P'$ be a partition of $[a,b]$ obtained by adding $r$ points to a", + "partition $P=\\{x_0,x_1, \\dots,x_n\\}$ of $[a,b].$ Then", + "\\begin{eqnarray}", + "S(P)\\ge S(P')\\ar\\ge S(P)-2Mr\\|P\\|\\label{eq:3.2.2}\\\\", + "\\arraytext{and}\\nonumber\\\\", + "s(P)\\le s(P')\\ar\\le s(P)+2Mr\\|P\\|\\label{eq:3.2.3}.", + "\\end{eqnarray}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will prove \\eqref{eq:3.2.2} and leave the proof of \\eqref{eq:3.2.3}", + "to you (Exercise~\\ref{exer:3.2.1}).", + "First suppose that $r=1$, so", + " $P'$ is obtained by adding one point $c$ to the", + "partition", + "$P=\\{x_0,x_1, \\dots,x_n\\}$; then", + "$x_{i-1}1$ and $P'$ is obtained by adding points $c_1$,", + "$c_2$, \\dots, $c_r$ to $P$. Let $P^{(0)}=P$ and, for $j\\ge1$, let", + "$P^{(j)}$ be the partition of $[a,b]$ obtained by adding $c_j$", + "to $P^{(j-1)}$. Then the result just proved implies that", + "$$", + "0\\le S(P^{(j-1)})-S(P^{(j)})\\le2M\\|P^{(j-1)}\\|,\\quad 1\\le j\\le r.", + "$$", + "\\newpage", + "\\noindent", + "Adding these inequalities and taking account of cancellations", + " yields", + "\\begin{equation} \\label{eq:3.2.5}", + "0\\le", + "S(P^{(0)})-S(P^{(r)})\\le2M(\\|P^{(0)}\\|+\\|P^{(1)}\\|+\\cdots+\\|P^{(r-1)}\\|).", + "\\end{equation}", + "Since $P^{(0)}=P$, $P^{(r)}=P'$, and $\\|P^{(k)}\\|\\le\\|P^{(k-1)}\\|$", + "for $1\\le k\\le r-1$, \\eqref{eq:3.2.5} implies that", + "$$", + "0\\le S(P)-S(P')\\le 2Mr\\|P\\|,", + "$$", + "which is equivalent to \\eqref{eq:3.2.2}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 247, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.2.4", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on $[a,b]$ and", + " $\\epsilon>0,$ there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:3.2.12}", + "\\overline{\\int_a^b}f(x)\\,dx\\le", + "S(P)<\\overline{\\int_a^b}f(x)\\,dx+\\epsilon", + "\\end{equation}", + "and", + "$$", + "\\underline{\\int_a^b} f(x)\\,dx\\ge s(P)>\\underline{\\int_a^b}", + "f(x)\\,dx-\\epsilon", + "$$", + "if $\\|P\\|<\\delta$." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We show that \\eqref{eq:3.2.12} holds if $\\|P\\|$ is sufficiently", + "small, and leave the rest of the proof to you (Exercise~\\ref{exer:3.2.3}).", + "The first inequality in \\eqref{eq:3.2.12} follows immediately from", + "Definition~\\ref{thmtype:3.1.3}.", + " To establish the second inequality,", + "suppose that $|f(x)|\\le K$ if $a\\le x\\le b$. From", + "Definition~\\ref{thmtype:3.1.3}, there is a partition $P_0=", + "\\{x_0,x_1, \\dots,x_{r+1}\\}$ of $[a,b]$ such that", + "\\begin{equation} \\label{eq:3.2.13}", + "S(P_0)<\\overline{\\int_a^b}f(x)\\,dx+\\frac{\\epsilon}{2}.", + "\\end{equation}", + "If $P$ is any partition of $[a,b]$, let $P'$ be constructed from the", + "partition points of $P_0$ and $P$. Then", + "\\begin{equation} \\label{eq:3.2.14}", + "S(P')\\le S(P_0),", + "\\end{equation}", + "by Lemma~\\ref{thmtype:3.2.1}. Since $P'$ is obtained by adding at most", + "$r$ points to $P$, Lemma~\\ref{thmtype:3.2.1} implies that", + "\\begin{equation} \\label{eq:3.2.15}", + "S(P')\\ge S(P)-2Kr\\|P\\|.", + "\\end{equation}", + " Now \\eqref{eq:3.2.13}, \\eqref{eq:3.2.14}, and \\eqref{eq:3.2.15}", + "imply that", + "\\begin{eqnarray*}", + "S(P)\\ar\\le S(P')+2Kr\\|P\\|\\\\", + "\\ar\\le S(P_0)+2Kr\\|P\\|\\\\", + "&<&\\overline{\\int_a^b} f(x)\\,dx+\\frac{\\epsilon}{2}+2Kr\\|P\\|.", + "\\end{eqnarray*}", + " Therefore, \\eqref{eq:3.2.12} holds if", + "$$", + "\\|P\\|<\\delta=\\frac{\\epsilon}{4Kr}.", + "$$", + "\\vskip-4.5ex" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.1", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.1" + ], + "ref_ids": [ + 316, + 316, + 246, + 246 + ] + } + ], + "ref_ids": [] + }, + { + "id": 248, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.5.3", + "categories": [], + "title": "", + "contents": [ + "If $w_f(x)<\\epsilon$ for $a\\le x \\le b,$ then there is a $\\delta>0$", + "such", + "that $W_f[a_1,b_1]\\le\\epsilon,$ provided that $[a_1,b_1]\\subset", + "[a,b]$ and", + "$b_1-a_1<\\delta.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We use the Heine--Borel theorem (Theorem~\\ref{thmtype:1.3.7}).", + "If $w_f(x)<\\epsilon$, there is an $h_x>0$ such that", + "\\begin{equation} \\label{eq:3.5.1}", + "|f(x')-f(x'')|<\\epsilon", + "\\end{equation}", + "\\newpage", + "\\noindent", + "if", + "\\begin{equation} \\label{eq:3.5.2}", + "x-2h_x0$, there is an $\\overline{x}$ from $E_\\rho$ in", + "$(x_0-h,x_0+h)$.", + "Since $[\\overline{x}-h_1,\\overline{x}+h_1] \\subset [x_0-h,x_0+h]$ for", + "sufficiently small $h_1$ and", + " $W_f[\\overline{x}-h_1,\\overline{x}+h_1]\\ge\\rho$, it follows that", + " $W_f[x_0-h,x_0+h]\\ge\\rho$ for all", + "$h>0$. This implies that $x_0\\in E_\\rho$, so $E_\\rho$ is closed", + "(Corollary~\\ref{thmtype:1.3.6}).", + "Now we will show that the stated condition in necessary for", + "integrability.", + "Suppose that the condition is not satisfied; that is, there is a", + "$\\rho>0$ and a $\\delta>0$ such that", + "$$", + "\\sum_{j=1}^p L(I_j)\\ge\\delta", + "$$", + "\\newpage", + "\\noindent", + "for every finite set $\\{I_1,I_2, \\dots, I_p\\}$ of open intervals", + "covering", + "$E_\\rho$. If", + "$P=", + "\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, then", + "\\begin{equation} \\label{eq:3.5.4}", + "S(P)-s(P)=\\sum_{j\\in A} (M_j-m_j)(x_j-x_{j-1})+\\sum_{j\\in B}", + "(M_j-m_j)(x_j-x_{j-1}),", + "\\end{equation}", + "where", + "$$", + "A=\\set{j}{[x_{j-1},x_j]\\cap E_\\rho\\ne\\emptyset}\\mbox{\\quad", + "and\\quad}", + "B=\\set{j}{[x_{j-1},x_j]\\cap E_\\rho=\\emptyset}\\negthickspace.", + "$$", + "Since $\\bigcup_{j\\in A} (x_{j-1},x_j)$ contains all points of $E_\\rho$", + "except any of $x_0$, $x_1$, \\dots, $x_n$ that may be in $E_\\rho$, and", + "each of", + "these finitely many possible exceptions can be covered by an open interval", + "of length as small as we please, our assumption on $E_\\rho$ implies that", + "$$", + "\\sum_{j\\in A} (x_j-x_{j-1})\\ge\\delta.", + "$$", + "Moreover, if $j\\in A$, then", + "$$", + "M_j-m_j\\ge\\rho,", + "$$", + "so \\eqref{eq:3.5.4} implies that", + "$$", + "S(P)-s(P)\\ge\\rho\\sum_{j\\in A} (x_j-x_{j-1})\\ge\\rho\\delta.", + "$$", + "Since this holds for every partition of $[a,b]$, $f$ is not integrable on", + "$[a,b]$, by Theorem~\\ref{thmtype:3.2.7}. This proves that the stated condition is", + "necessary for integrability.", + "For sufficiency, let $\\rho$ and $\\delta$ be positive numbers and let", + "$I_1$, $I_2$, \\dots, $I_p$ be open intervals that cover $E_\\rho$ and", + "satisfy", + "\\eqref{eq:3.5.3}. Let", + "$$", + "\\widetilde{I}_j=[a,b]\\cap\\overline{I}_j.", + "$$", + "($\\overline{I}_j=\\mbox{closure of } I$.) After combining any of", + "$\\widetilde{I}_1$, $\\widetilde{I}_2$, \\dots, $\\widetilde{I}_p$ that overlap, we", + "obtain a set of pairwise disjoint closed subintervals", + "$$", + "C_j=[\\alpha_j,\\beta_j],\\quad 1\\le j\\le q\\ (\\le p),", + "$$", + "of $[a,b]$ such that", + "\\begin{equation} \\label{eq:3.5.5}", + "a\\le\\alpha_1<\\beta_1<\\alpha_2<\\beta_2\\cdots<", + "\\alpha_{q-1}<\\beta_{q-1}<\\alpha_q<\\beta_q\\le b,", + "\\end{equation}", + "\\begin{equation} \\label{eq:3.5.6}", + "\\sum_{i=1}^q\\, (\\beta_i-\\alpha_i)<\\delta", + "\\end{equation}", + "and", + "$$", + "w_f(x)<\\rho,\\quad\\beta_j\\le x\\le\\alpha_{j+1},\\quad 1\\le j\\le q-1.", + "$$", + "Also, $w_f(x)<\\rho$ for $a\\le x\\le\\alpha_1$ if $a<\\alpha_1$ and for", + "$\\beta_q\\le x\\le b$ if $\\beta_q0$, let", + "$$", + "\\delta=\\frac{\\epsilon}{4K}\\mbox{\\quad", + "and\\quad}\\rho=\\frac{\\epsilon}{", + "2(b-a)}.", + "$$", + "Then \\eqref{eq:3.5.7} yields", + "$$", + "S(P)-s(P)<\\epsilon,", + "$$", + "and Theorem~\\ref{thmtype:3.2.7} implies that $f$ is", + "integrable on $[a,b]$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.3.6", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:3.5.3", + "TRENCH_REAL_ANALYSIS-thmtype:3.2.7" + ], + "ref_ids": [ + 274, + 50, + 248, + 50 + ] + } + ], + "ref_ids": [] + }, + { + "id": 250, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.4", + "categories": [], + "title": "", + "contents": [ + "Suppose that for $n$ sufficiently large", + " $($that is$,$ for $n \\ge\\mbox{some", + "integer }N$$)$", + " the terms of", + "$\\sum_{n=k}^\\infty a_n$ satisfy", + " some condition that implies convergence", + "of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$", + "converges$.$", + "Similarly, suppose that for $n$ sufficiently large the terms", + "$\\sum_{n=k}^\\infty a_n$ satisfy", + " some condition that implies divergence", + "of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$", + "diverges$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "In terms of the partial sums $\\{A_n\\}$ of $\\sum a_n$,", + "$$", + "a_n+a_{n+1}+\\cdots+a_m=A_m-A_{n-1}.", + "$$", + "Therefore, \\eqref{eq:4.3.3} can be written as", + "$$", + "|A_m-A_{n-1}|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N.", + "$$", + "Since $\\sum a_n$ converges if and only if $\\{A_n\\}$ converges,", + "Theorem~\\ref{thmtype:4.1.13} implies the conclusion." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" + ], + "ref_ids": [ + 89 + ] + } + ], + "ref_ids": [] + }, + { + "id": 251, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", + "categories": [], + "title": "", + "contents": [ + "If $g$ and $h$ are defined on $S,$ then", + "\\begin{eqnarray*}", + "\\|g+h\\|_S\\ar\\le\\|g\\|_S+\\|h\\|_S\\\\", + "\\arraytext{and}\\\\", + "\\|gh\\|_S\\ar\\le\\|g\\|_S\\|h\\|_S.", + "\\end{eqnarray*}", + "Moroever$,$ if either $g$ or $h$ is bounded on $S,$ then", + "$$", + "\\|g-h\\|_S\\ge\\left|\\|g\\|_S-\\|h\\|_S\\|\\right|.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "For necessity, suppose that $\\{F_n\\}$ converges uniformly to", + "$F$ on $S$. Then, if $\\epsilon>0$, there is an integer $N$ such that", + "$$", + "\\|F_k-F\\|_S<\\frac{\\epsilon}{2}\\mbox{\\quad if\\quad} k\\ge N.", + "$$", + "Therefore,", + "\\begin{eqnarray*}", + "\\|F_n-F_m\\|_S\\ar=\\|(F_n-F)+(F-F_m)\\|_S\\\\", + "\\ar\\le \\|F_n-F\\|_S+\\|F-F_m\\|_S \\mbox{\\quad", + "(Lemma~\\ref{thmtype:4.4.2})\\quad}\\\\", + "&<&\\frac{\\epsilon}{2}+\\frac{\\epsilon}{2}=\\epsilon\\mbox{\\quad if\\quad}", + "m, n\\ge N.", + "\\end{eqnarray*}", + "For sufficiency, we first observe that \\eqref{eq:4.4.2} implies that", + "$$", + "|F_n(x)-F_m(x)|<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N,", + "$$", + "for any fixed $x$ in $S$. Therefore, Cauchy's convergence criterion", + "for sequences of constants (Theorem~\\ref{thmtype:4.1.13})", + "implies that", + "$\\{F_n(x)\\}$ converges for each $x$ in $S$; that is, $\\{F_n\\}$", + "converges pointwise to a limit function $F$ on $S$. To see that the", + "convergence is uniform, we write", + "\\begin{eqnarray*}", + "|F_m(x)-F(x) |\\ar=|[F_m(x)-F_n(x)]+[F_n(x)-F(x)]|\\\\", + "\\ar\\le |F_m(x)-F_n(x)|+| F_n(x)-F(x)|\\\\", + "\\ar\\le \\|F_m-F_n\\|_S+|F_n(x)-F(x)|.", + "\\end{eqnarray*}", + "This and \\eqref{eq:4.4.2} imply that", + "\\begin{equation} \\label{eq:4.4.3}", + "|F_m(x)-F(x)|<\\epsilon+|F_n(x)-F(x)|\\quad\\mbox {if}\\quad n, m\\ge N.", + "\\end{equation}", + "Since $\\lim_{n\\to\\infty}F_n(x)=F(x)$,", + "$$", + "|F_n(x)-F(x)|<\\epsilon", + "$$", + "for some $n\\ge N$, so \\eqref{eq:4.4.3} implies that", + "$$", + "|F_m(x)-F(x)|<2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", + "$$", + "But this inequality holds for all $x$ in $S$, so", + "$$", + "\\|F_m-F\\|_S\\le2\\epsilon\\mbox{\\quad if\\quad} m\\ge N.", + "$$", + "Since $\\epsilon$ is an arbitrary positive number, this implies that", + "$\\{F_n\\}$ converges uniformly to $F$ on~$S$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", + "TRENCH_REAL_ANALYSIS-thmtype:4.1.13" + ], + "ref_ids": [ + 251, + 89 + ] + } + ], + "ref_ids": [] + }, + { + "id": 252, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.5", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{X}$ and $\\mathbf{Y}$ are any two vectors in $\\R^n,$ then", + "\\begin{equation} \\label{eq:5.1.3}", + "|\\mathbf{X}\\cdot\\mathbf{Y}|\\le |\\mathbf{X}|\\,|\\mathbf{Y}|,", + "\\end{equation}", + "with equality if and only if one of the vectors is a scalar", + "multiple of the other$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $\\mathbf{Y}=\\mathbf{0}$, then both sides", + "of \\eqref{eq:5.1.3} are $\\mathbf{0}$, so \\eqref{eq:5.1.3} holds, with equality.", + "In this case, $\\mathbf{Y}=0\\mathbf{X}$.", + "Now suppose that $\\mathbf{Y}\\ne\\mathbf{0}$ and", + " $t$ is any real number. Then", + "\\begin{equation}\\label{eq:5.1.4}", + "\\begin{array}{rcl}", + "0\\ar\\le \\dst{\\sum^n_{i=1} (x_i-ty_i)^2}\\\\", + "\\ar=\\dst{\\sum^n_{i=1} x^2_i-2t\\sum^n_{i=1} x_iy_i+t^2\\sum^n_{i=1}", + "y^2_i}\\\\\\\\", + "\\ar=|\\mathbf{X}|^2-2(\\mathbf{X}\\cdot\\mathbf{Y})t+t^2|\\mathbf{Y}|^2.", + "\\end{array}", + "\\end{equation}", + "The last expression is a second-degree polynomial $p$", + "in $t$. From the quadratic formula, the zeros of $p$ are", + "$$", + "t=\\frac{(\\mathbf{X}\\cdot\\mathbf{Y})\\pm\\sqrt{(\\mathbf{X}\\cdot\\mathbf{Y})^2-", + "|\\mathbf{X}|^2|\\mathbf{Y}|^2}}{ |\\mathbf{Y}|^2}.", + "$$", + "Hence,", + "\\begin{equation}\\label{eq:5.1.5}", + "(\\mathbf{X}\\cdot\\mathbf{Y})^2\\le |\\mathbf{X}|^2|\\mathbf{Y}|^2,", + "\\end{equation}", + "because if not, then $p$ would have two distinct real zeros and", + "therefore", + "be negative between them (Figure~\\ref{figure:5.1.1}), contradicting the", + "inequality \\eqref{eq:5.1.4}. Taking square roots in \\eqref{eq:5.1.5} yields", + "\\eqref{eq:5.1.3} if $\\mathbf{Y}\\ne\\mathbf{0}$.", + "If $\\mathbf{X}=t\\mathbf{Y}$, then", + "$|\\mathbf{X}\\cdot\\mathbf{Y}|=|\\mathbf{X}||\\mathbf{Y}|", + "=|t||\\mathbf{Y}|^2$ (verify), so equality holds in \\eqref{eq:5.1.3}.", + "Conversely, if equality holds in \\eqref{eq:5.1.3}, then $p$ has the real", + "zero $t_0=(\\mathbf{X}\\cdot\\mathbf{Y})/|\\mathbf{Y}\\|^2$, and", + "$$", + "\\sum_{i=1}^n(x_i-t_0y_i)^2=0", + "$$", + "\\nopagebreak", + "from \\eqref{eq:5.1.4}; therefore, $\\mathbf{X}=t_0\\mathbf{Y}$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 253, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.12", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{X}_1$ and $\\mathbf{X}_2$ are in $S_r(\\mathbf{X}_0)$ for some $r>0$,", + "then so is every point on", + "the line segment from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "The line segment is given by", + "$$", + "\\mathbf{X}=t\\mathbf{X}_2+(1-t)\\mathbf{X}_1,\\quad 00$. If", + "$$", + "|\\mathbf{X}_1-\\mathbf{X}_0|0$. Our assumptions imply that there is", + "a $\\delta>0$ such that $f_{x_1}, f_{x_2}, \\dots, f_{x_n}$ are defined", + "in the $n$-ball", + "$$", + "S_\\delta (\\mathbf{X}_0)=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\delta}", + "$$", + "and", + "\\begin{equation}\\label{eq:5.3.24}", + "|f_{x_j}(\\mathbf{X})-f_{x_j}(\\mathbf{X}_0)|<\\epsilon\\mbox{\\quad if\\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta,\\quad 1\\le j\\le n.", + "\\end{equation}", + "Let $\\mathbf{X}=(x_1,x_, \\dots,x_n)$ be in $S_\\delta(\\mathbf{X}_0)$.", + "Define", + "$$", + "\\mathbf{X}_j=(x_1, \\dots,x_j, x_{j+1,0}, \\dots,x_{n0}),\\quad 1\\le j\\le n-1,", + "$$", + "and", + "$\\mathbf{X}_n=\\mathbf{X}$.", + "Thus, for $1\\le j\\le n$, $\\mathbf{X}_j$ differs from $\\mathbf{X}_{j-1}$", + " in the", + "$j$th component only, and the line segment from $\\mathbf{X}_{j-1}$ to", + "$\\mathbf{X}_j$ is in $S_\\delta (\\mathbf{X}_0)$.", + "Now write", + "\\begin{equation}\\label{eq:5.3.25}", + "f(\\mathbf{X})-f(\\mathbf{X}_0)=f(\\mathbf{X}_n)-f(\\mathbf{X}_0)=", + "\\sum^n_{j=1}\\,[f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})],", + "\\end{equation}", + "and consider the auxiliary functions", + "\\begin{equation}\\label{eq:5.3.26}", + "\\begin{array}{rcl}", + "g_1(t)\\ar=f(t,x_{20}, \\dots,x_{n0}),\\\\[2\\jot]", + "g_j(t)\\ar=f(x_1, \\dots,x_{j-1},t,x_{j+1,0}, \\dots,x_{n0}),\\quad 2\\le j\\le", + "n-1,\\\\[2\\jot]", + "g_n(t)\\ar=f(x_1, \\dots,x_{n-1},t),", + "\\end{array}", + "\\end{equation}", + "where, in each case, all variables except $t$ are temporarily regarded", + "as constants. Since", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g_j(x_j)-g_j(x_{j0}),", + "$$", + "the mean value theorem implies that", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=g'_j(\\tau_j)(x_j-x_{j0}),", + "$$", + "\\newpage", + "\\noindent", + "where $\\tau_j$ is between $x_j$ and $x_{j0}$. From \\eqref{eq:5.3.26},", + "$$", + "g'_j(\\tau_j)=f_{x_j}(\\widehat{\\mathbf{X}}_j),", + "$$", + "where $\\widehat{\\mathbf{X}}_j$ is on the line segment from $\\mathbf{X}_{j-1}$ to", + "$\\mathbf{X}_j$. Therefore,", + "$$", + "f(\\mathbf{X}_j)-f(\\mathbf{X}_{j-1})=f_{x_j}(\\widehat{\\mathbf{X}}_j)(x_j-x_{j0}),", + "$$", + "and \\eqref{eq:5.3.25} implies that", + "\\begin{eqnarray*}", + "f(\\mathbf{X})-f(\\mathbf{X}_0)\\ar=\\sum^n_{j=1} f_{x_j} (\\widehat{\\mathbf{X}}_j)(x_j-x_{j0})\\\\", + "\\ar=\\sum^n_{j=1} f_{x_j}(\\mathbf{X}_0) (x_j-x_{j0})+\\sum^n_{j=1}", + "\\,[f_{x_j}(\\widehat{\\mathbf{X}}_j)-f_{x_j}(\\mathbf{X}_0)](x_j-x_{j0}).", + "\\end{eqnarray*}", + "From this and \\eqref{eq:5.3.24},", + "$$", + "\\left|f(\\mathbf{X})-f(\\mathbf{X}_0)-\\sum^n_{j=1}", + "f_{x_j}(\\mathbf{X}_{0})", + "(x_j-x_{j0})\\right|\\le", + "\\epsilon\\sum^n_{j=1} |x_j-x_{j0}|\\le n\\epsilon |\\mathbf{X}-\\mathbf{X}_0|,", + "$$", + "which implies that $f$ is differentiable at $\\mathbf{X}_0$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 255, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.2", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is differentiable at", + "$$", + " \\mathbf{U}_0=(u_{10}, u_{20}, \\dots,u_{m0}),", + "$$", + " and", + " define", + "$$", + "M=\\left(\\sum_{i=1}^n\\sum_{j=1}^m\\left(\\frac{\\partial g_i(\\mathbf{U}_0}", + "{\\partial u_j}\\right)^2\\right)^{1/2}.", + "$$", + "Then$,$ if $\\epsilon>0,$ there is a $\\delta>0$ such that", + "$$", + "\\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|}", + "{|\\mathbf{U}-\\mathbf{U}_{0}|}", + "0,$ there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:6.2.8}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|<", + "(\\|\\mathbf{F}'(\\mathbf{X}_{0})\\|", + "+\\epsilon) |\\mathbf{X}-\\mathbf{Y}|", + "\\mbox{\\quad if\\quad}\\mathbf{A},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Consider the auxiliary function", + "\\begin{equation} \\label{eq:6.2.9}", + "\\mathbf{G}(\\mathbf{X})=\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}_0)\\mathbf{X}.", + "\\end{equation}", + "The components of $\\mathbf{G}$ are", + "$$", + "g_i(\\mathbf{X})=f_i(\\mathbf{X})-\\sum_{j=1}^n", + "\\frac{\\partial f_i(\\mathbf{X}_{0})", + "\\partial x_j} x_j,", + "$$", + "so", + "$$", + "\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}=", + "\\frac{\\partial f_i(\\mathbf{X})}", + "{\\partial x_j}-\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}.", + "$$", + "\\newpage", + "\\noindent", + "Thus, $\\partial g_i/\\partial x_j$ is continuous on $N$ and zero at", + "$\\mathbf{X}_0$. Therefore, there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:6.2.10}", + "\\left|\\frac{\\partial g_i(\\mathbf{X})}{\\partial x_j}\\right|<\\frac{\\epsilon}{", + "\\sqrt{mn}}\\mbox{\\quad for \\quad}1\\le i\\le m,\\quad 1\\le j\\le n,", + "\\mbox{\\quad if \\quad}", + "|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", + "\\end{equation}", + "Now suppose that $\\mathbf{X}$, $\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0)$. By", + "Theorem~\\ref{thmtype:5.4.5},", + "\\begin{equation}\\label{eq:6.2.11}", + "g_i(\\mathbf{X})-g_i(\\mathbf{Y})=\\sum_{j=1}^n", + "\\frac{\\partial g_i(\\mathbf{X}_i)}{\\partial x_j}(x_j-y_j),", + "\\end{equation}", + "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$,", + "so $\\mathbf{X}_i\\in B_\\delta(\\mathbf{X}_0)$. From \\eqref{eq:6.2.10},", + "\\eqref{eq:6.2.11}, and Schwarz's inequality,", + "$$", + "(g_i(\\mathbf{X})-g_i(\\mathbf{Y}))^2\\le\\left(\\sum_{j=1}^n\\left[\\frac{\\partial", + "g_i", + "(\\mathbf{X}_i)}{\\partial x_j}\\right]^2\\right)", + "|\\mathbf{X}-\\mathbf{Y}|^2", + "<\\frac{\\epsilon^2}{ m} |\\mathbf{X}-\\mathbf{Y}|^2.", + "$$", + "Summing this from $i=1$ to $i=m$ and taking square roots yields", + "\\begin{equation}\\label{eq:6.2.12}", + "|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|<\\epsilon", + "|\\mathbf{X}-\\mathbf{Y}|", + "\\mbox{\\quad if\\quad}\\mathbf{X}, \\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", + "\\end{equation}", + "To complete the proof, we note that", + "\\begin{equation}\\label{eq:6.2.13}", + "\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})=", + "\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})+\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y}),", + "\\end{equation}", + " so \\eqref{eq:6.2.12} and the triangle inequality imply \\eqref{eq:6.2.8}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.5" + ], + "ref_ids": [ + 164 + ] + } + ], + "ref_ids": [] + }, + { + "id": 257, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.6", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{F}:\\R^n\\to\\R^n$ is continuously", + "differentiable on a neighborhood of $\\mathbf{X}_0$", + " and $\\mathbf{F}'(\\mathbf{X}_0)$ is nonsingular$.$ Let", + "\\begin{equation}\\label{eq:6.2.14}", + "r=\\frac{1}{\\|(\\mathbf{F}'(\\mathbf{X}_0))^{-1}\\|}.", + "\\end{equation}", + "Then$,$ for every $\\epsilon>0,$ there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:6.2.15}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|\\ge (r-\\epsilon)", + "|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad} \\mathbf{X},\\mathbf{Y}\\in", + "B_\\delta(\\mathbf{X}_{0}).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $\\mathbf{X}$ and $\\mathbf{Y}$ be arbitrary points in", + "$D_\\mathbf{F}$ and let $\\mathbf{G}$ be as in \\eqref{eq:6.2.9}. From", + "\\eqref{eq:6.2.13},", + "\\begin{equation} \\label{eq:6.2.16}", + "|\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|\\ge\\big|", + "|\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}", + "-\\mathbf{Y})|-|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{Y})|\\big|,", + "\\end{equation}", + "Since", + "$$", + "\\mathbf{X}-\\mathbf{Y}=[\\mathbf{F}'(\\mathbf{X}_0)]^{-1}", + "\\mathbf{F}'(\\mathbf{X}_{0})", + "(\\mathbf{X}-\\mathbf{Y}),", + "$$", + "\\eqref{eq:6.2.14} implies that", + "$$", + "|\\mathbf{X}-\\mathbf{Y}|\\le \\frac{1}{ r} |\\mathbf{F}'(\\mathbf{X}_0)", + "(\\mathbf{X}-\\mathbf{Y}|,", + "$$", + "so", + "\\begin{equation}\\label{eq:6.2.17}", + "|\\mathbf{F}'(\\mathbf{X}_0)(\\mathbf{X}-\\mathbf{Y})|\\ge r|\\mathbf{X}-\\mathbf{Y}|.", + "\\end{equation}", + " Now choose $\\delta>0$ so that \\eqref{eq:6.2.12} holds.", + "Then \\eqref{eq:6.2.16} and \\eqref{eq:6.2.17} imply \\eqref{eq:6.2.15}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 258, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.2.7", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable", + "on an open set containing a compact set $D,$ then there is a constant", + "$M$ such that", + "\\begin{equation}\\label{eq:6.2.18}", + "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}|", + "\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "On", + "$$", + "S=\\set{(\\mathbf{X},\\mathbf{Y})}{\\mathbf{X},\\mathbf{Y}\\in D}\\subset \\R^{2n}", + "$$", + "define", + "$$", + "g(\\mathbf{X},\\mathbf{Y})=\\left\\{\\casespace\\begin{array}{ll}", + "\\dst{\\frac{|\\mathbf{F}(\\mathbf{Y})-", + "\\mathbf{F}(\\mathbf{X})", + "-\\mathbf{F}'(\\mathbf{X})(\\mathbf{Y}-\\mathbf{X})|}{ |\\mathbf{Y}-\\mathbf{X}|}},&", + "\\mathbf{Y}\\ne\\mathbf{X},\\\\[2\\jot]", + " 0,&\\mathbf{Y}=\\mathbf{X}.\\end{array}\\right.", + "$$", + "Then $g$ is continuous for all $(\\mathbf{X},\\mathbf{Y})$ in $S$", + "such that $\\mathbf{X}\\ne \\mathbf{Y}$. We now show that if $\\mathbf{X}_0\\in D$,", + "then", + "\\begin{equation}\\label{eq:6.2.19}", + "\\lim_{(\\mathbf{X},\\mathbf{Y})\\to (\\mathbf{X}_0,\\mathbf{X}_0)}", + "g(\\mathbf{X},\\mathbf{Y})=0", + "=g(\\mathbf{X}_0,\\mathbf{X}_0);", + "\\end{equation}", + "that is, $g$ is also continuous at points $(\\mathbf{X}_0,\\mathbf{X}_0)$ in", + "$S$.", + "Suppose that $\\epsilon>0$ and $\\mathbf{X}_0\\in D$. Since the partial", + "derivatives of $f_1$, $f_2$, \\dots, $f_m$ are continuous on an open", + "set containing $D$, there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:6.2.20}", + "\\left|\\frac{\\partial f_i(\\mathbf{Y})}{\\partial x_j}-\\frac{\\partial", + "f_i(\\mathbf{X})", + "}{\\partial x_j}\\right|<\\frac{\\epsilon}{\\sqrt{mn}}\\mbox{\\quad if\\quad}", + "\\mathbf{X},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0),\\ 1\\le i\\le m,\\", + "1\\le j\\le n.", + "\\end{equation}", + "(Note that $\\partial f_i/\\partial x_j$ is uniformly continuous on", + "$\\overline{B_\\delta(\\mathbf{X}_0)}$ for $\\delta$ sufficiently small, from", + "Theorem~\\ref{thmtype:5.2.14}.) Applying", + "Theorem~\\ref{thmtype:5.4.5}", + "to $f_1$, $f_2$, \\dots, $f_m$, we find that if $\\mathbf{X}$, $\\mathbf{Y}\\in", + "B_\\delta", + "(\\mathbf{X}_0)$, then", + "$$", + "f_i(\\mathbf{Y})-f_i(\\mathbf{X})=\\sum_{j=1}^n", + "\\frac{\\partial f_i(\\mathbf{X}_{i})}", + "{\\partial x_j} (y_j-x_j),", + "$$", + "where $\\mathbf{X}_i$ is on the line segment from $\\mathbf{X}$ to $\\mathbf{Y}$.", + "From this,", + "\\begin{eqnarray*}", + "\\left[f_i(\\mathbf{Y})-f_i(\\mathbf{X})", + "-\\dst{\\sum_{j=1}^n}", + "\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j} (y_j-x_j)\\right]^2", + "\\ar=\\left[\\sum_{j=1}^n\\left[\\frac{\\partial f_i(\\mathbf{X}_i)}{\\partial", + "x_j}-", + "\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j}\\right] (y_j-x_j)\\right]^2\\\\", + "\\ar\\le |\\mathbf{Y}-\\mathbf{X}|^2\\sum_{j=1}^n", + "\\left[\\frac{\\partial f_i(\\mathbf{X}_{i})}", + "{\\partial x_j}", + "-\\frac{\\partial f_i(\\mathbf{X})}{\\partial x_j}\\right]^2\\\\", + "\\ar{}\\mbox{(by Schwarz's inequality)}\\\\", + "\\ar< \\frac{\\epsilon^2}{ m} |\\mathbf{Y}-\\mathbf{X}|^2\\quad\\mbox{\\quad (by", + "\\eqref{eq:6.2.20})\\quad}.", + "\\end{eqnarray*}", + "Summing from $i=1$ to $i=m$ and taking square roots yields", + "$$", + "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X})", + "(\\mathbf{Y}-\\mathbf{X})|", + "<\\epsilon |\\mathbf{Y}-\\mathbf{X}|\\mbox{\\quad if\\quad}", + "\\mathbf{X},\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_0).", + "$$", + "\\nopagebreak", + "This implies \\eqref{eq:6.2.19} and completes the proof that $g$ is", + "continuous on $S$.", + "\\newpage", + " Since $D$ is compact, so is $S$", + "(Exercise~\\ref{exer:5.1.27}).", + "Therefore, $g$ is bounded on $S$", + "(Theorem~\\ref{thmtype:5.2.12}); thus, for some $M_1$,", + "$$", + "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X}) (\\mathbf{Y}", + "-\\mathbf{X})|\\le M_1|\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad}", + "\\mathbf{X},\\mathbf{Y}\\in D.", + "$$", + "But", + "\\begin{equation}\\label{eq:6.2.21}", + "\\begin{array}{rcl}", + "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X}) |\\ar\\le", + "|\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})-\\mathbf{F}'(\\mathbf{X})", + "(\\mathbf{Y}-\\mathbf{X})|+|\\mathbf{F}'(\\mathbf{X})(\\mathbf{Y}-\\mathbf{X})|\\\\", + "\\ar\\le (M_1+\\|\\mathbf{F}'(\\mathbf{X})\\|) |(\\mathbf{Y}-\\mathbf{X}|.", + "\\end{array}", + "\\end{equation}", + "Since", + "$$", + "\\|\\mathbf{F}'(\\mathbf{X})\\|", + "\\le\\left(\\sum_{i=1}^m\\sum_{j=1}^n\\left[\\frac{\\partial", + "f_i(\\mathbf{X}) }{\\partial x_j}\\right]^2\\right)^{1/2}", + "$$", + "and the partial derivatives $\\{\\partial f_i/\\partial x_j\\}$ are", + "bounded on $D$, it follows that $\\|\\mathbf{F}'(\\mathbf{X})\\|$ is bounded on", + "$D$; that is, there is a constant $M_2$ such that", + "$$", + "\\|\\mathbf{F}'(\\mathbf{X})\\|\\le M_2,\\quad\\mathbf{X}\\in D.", + "$$", + "Now \\eqref{eq:6.2.21} implies \\eqref{eq:6.2.18} with $M=M_1+M_2$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", + "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", + "TRENCH_REAL_ANALYSIS-thmtype:5.2.12" + ], + "ref_ids": [ + 154, + 164, + 152 + ] + } + ], + "ref_ids": [] + }, + { + "id": 259, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.6", + "categories": [], + "title": "", + "contents": [ + "Suppose that $|f(\\mathbf{X})|\\le", + "M$ if $\\mathbf{X}$ is in the rectangle", + "$$", + "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n].", + "$$", + "Let ${\\bf P}=P_1\\times P_2\\times\\cdots\\times P_n$ and ${\\bf P}'=", + "P_1'\\times P_2'\\times\\cdots\\times P_n'$ be partitions of $R,$ where", + "$P_j'$ is obtained by adding $r_j$ partition points to $P_j,$", + "$1\\le j\\le n.$ Then", + "\\begin{equation}\\label{eq:7.1.16}", + "S({\\bf P})\\ge S({\\bf P}')\\ge S({\\bf P})-2MV(R)\\left(\\sum_{j=1}^n", + "\\frac{r_j}{ b_j-a_j}\\right)\\|{\\bf P}\\|", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.1.17}", + "s({\\bf P})\\le s({\\bf P}')\\le s({\\bf P})+2MV(R)\\left(\\sum_{j=1}^n", + "\\frac{r_j", + "}{ b_j-a_j}\\right)\\|{\\bf P}\\|.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will prove", + " \\eqref{eq:7.1.16} and leave the proof of \\eqref{eq:7.1.17} to you", + "(Exercise~\\ref{exer:7.1.7}).", + "First suppose that", + " $P_1'$ is obtained by adding one point to $P_1$, and", + "$P_j'=P_j$ for $2\\le j\\le n$.", + "If $P_r$ is", + "defined by", + "$$", + "P_r: a_r=a_{r0}0,$ there is", + " a $\\delta>0$ such that", + "\\vspace{4pt}", + "$$", + "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le S({\\bf P})<\\overline{\\int_R}\\,", + "f(\\mathbf{X})\\,d\\mathbf{X}+\\epsilon", + "$$", + "\\vspace{4pt}", + "and", + "\\vspace{4pt}", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\ge s({\\bf P})>", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}-\\epsilon", + "$$", + "\\vspace{4pt}", + "if $\\|{\\bf P}\\|<\\delta.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Exercise~\\ref{exer:7.1.10}.", + "The next theorem is analogous to Theorem~3.2.5.", + "\\begin{theorem}\\label{thmtype:7.1.10}", + "If $f$ is bounded on a rectangle $R$ and", + "\\vspace{2pt}", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=L,", + "$$", + "\\vspace{2pt}", + "then $f$ is integrable on $R,$ and", + "\\vspace{2pt}", + "$$", + "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", + "$$", + "\\end{theorem}" + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 261, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.15", + "categories": [], + "title": "", + "contents": [ + "The union of finitely many sets with zero content has zero content$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Suppose that $\\epsilon>0$. Since $E$ has zero content, there are", + "rectangles", + "$T_1$, $T_2$, \\dots, $T_m$ such that", + "\\begin{equation} \\label{eq:7.1.31}", + "E\\subset\\bigcup_{j=1}^m T_j", + "\\end{equation}", + "and", + "\\begin{equation} \\label{eq:7.1.32}", + "\\sum_{j=1}^m V(T_j)<\\epsilon.", + "\\end{equation}", + " We may assume that", + "$T_1$, $T_2$, \\dots, $T_m$ are contained in $R$, since, if not, their", + "intersections with", + "$R$ would be contained in $R$, and still satisfy \\eqref{eq:7.1.31}", + "and \\eqref{eq:7.1.32}.", + " We may also assume that if $T$ is any rectangle such", + "that", + "\\begin{equation}\\label{eq:7.1.33}", + "T\\bigcap\\left(\\bigcup_{j=1}^m T_j^0\\right)=\\emptyset, \\mbox{\\quad", + "then", + "\\quad}T\\cap E=\\emptyset", + "\\end{equation}", + "\\newpage", + "\\noindent", + "since if this were not so, we could make it so by enlarging", + "$T_1$, $T_2$, \\dots, $T_m$", + "slightly while maintaining \\eqref{eq:7.1.32}. Now suppose that", + "\\vspace*{1pt}", + "$$", + "T_j=[a_{1j},b_{1j}]\\times [a_{2j},b_{2j}]\\times\\cdots\\times", + "[a_{nj},b_{nj}],\\quad 1\\le j\\le m,", + "$$", + "\\vspace*{1pt}", + "\\noindent let $P_{i0}$ be the partition of $[a_i,b_i]$ (see", + "\\eqref{eq:7.1.30}) with partition points", + "$$", + "a_i,b_i,a_{i1},b_{i1},a_{i2},b_{i2}, \\dots,a_{im},b_{im}", + "\\vspace*{1pt}", + "$$", + "(these are not in increasing order), $1\\le i\\le n$, and let", + "\\vspace*{1pt}", + "$$", + "{\\bf P}_0=P_{10}\\times P_{20}\\times\\cdots\\times P_{n0}.", + "$$", + "\\vspace*{1pt}", + "\\noindent\\hskip-.3em Then ${\\bf P}_0$ consists of rectangles whose", + "union equals $\\cup_{j=1}^m T_j$", + "and other rectangles", + "$T'_1$, $T'_2$, \\dots, $T'_k$ that do not intersect $E$. (We need", + "\\eqref{eq:7.1.33} to be sure that $T'_i\\cap E=\\emptyset,", + "1\\le i\\le k.)$ If we let", + "$$", + "B=\\bigcup_{j=1}^m T_j\\mbox{\\quad and\\quad} C=\\bigcup^k_{i=1} T'_i,", + "$$", + "then $R=B\\cup C$ and $f$ is continuous on the compact set $C$.", + "If ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a refinement of ${\\bf P}_0$,", + "then every subrectangle $R_j$ of ${\\bf P}$ is contained entirely in", + "$B$ or entirely in $C$. Therefore, we can write", + "\\vspace*{1pt}", + "\\begin{equation}\\label{eq:7.1.34}", + "S({\\bf P})-s({\\bf P})=\\Sigma_1(M_j-m_j)", + "V(R_j)+\\Sigma_2(M_j-m_j)V(R_j),", + "\\end{equation}", + "\\vspace*{1pt}", + "\\noindent \\hskip-.3em", + "where $\\Sigma_1$ and $\\Sigma_2$ are summations over values of $j$ for", + "which $R_j\\subset B$ and $R_j\\subset C$, respectively. Now suppose that", + "$$", + "|f(\\mathbf{X})|\\le M\\mbox{\\quad for $\\mathbf{X}$ in $R$}.", + "$$", + "Then", + "\\begin{equation}\\label{eq:7.1.35}", + "\\Sigma_1(M_j-m_j) V(R_j)\\le2M\\,\\Sigma_1 V(R_j)=2M\\sum_{j=1}^m V(T_j)<", + "2M\\epsilon,", + "\\end{equation}", + "from \\eqref{eq:7.1.32}.", + "Since $f$ is uniformly continuous on the compact set $C$", + "(Theorem~\\ref{thmtype:5.2.14}),", + "there is a $\\delta>0$ such that $M_j-m_j<\\epsilon$ if", + "$\\|{\\bf P}\\|< \\delta$ and $R_j\\subset C$; hence,", + "$$", + "\\Sigma_2(M_j-m_j)V(R_j)<\\epsilon\\Sigma_2\\, V(R_j)\\le\\epsilon V(R).", + "$$", + "This, \\eqref{eq:7.1.34}, and \\eqref{eq:7.1.35} imply that", + "$$", + "S({\\bf P})-s({\\bf P})<[2M+V(R)]\\epsilon", + "$$", + "if $\\|{\\bf P}\\|<\\delta$ and ${\\bf P}$ is a refinement of ${\\bf P}_0$.", + "Therefore, Theorem~\\ref{thmtype:7.1.12} implies that $f$ is integrable on", + "$R$.", + "\\enlargethispage{4\\baselineskip}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.2.14", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.12" + ], + "ref_ids": [ + 154, + 195 + ] + } + ], + "ref_ids": [] + }, + { + "id": 262, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.29", + "categories": [], + "title": "", + "contents": [ + "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable", + "on $S.$ Then", + " $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and", + "$$", + "\\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Definition~\\ref{thmtype:7.1.17} with $f$ and $S$ replaced by $f_S$", + "and $T$,", + "\\pagebreak", + "$$", + "(f_S)_T(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f_S(\\mathbf{X}),&\\mathbf{X}\\in T,\\\\", + " 0,&\\mathbf{X}\\not\\in T.\\end{array}\\right.", + "$$", + " Since $S\\subset T$, $(f_S)_T=f_S$.", + "(Verify.) Now suppose that $R$ is a rectangle containing $T$.", + " Then $R$ also", + "contains $S$ (Figure~\\ref{figure:7.1.7}),", + " \\vspace*{12pt}", + " \\centereps{2.3in}{1.45in}{fig070107.eps}", + " \\vskip6pt", + " \\refstepcounter{figure}", + " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.1.7}", + " \\vskip12pt", + "\\noindent so", + "$$", + "\\begin{array}{rcll}", + "\\dst\\int_Sf(\\mathbf{X})\\,d\\mathbf{X}\\ar=\\dst\\int_Rf_S(\\mathbf{X})\\,d\\mathbf{X}&", + "\\mbox{(Definition~\\ref{thmtype:7.1.17}, applied to $f$ and $S$})\\\\[4\\jot]", + "\\ar=\\dst\\int_R(f_S)_T(\\mathbf{X})\\,d\\mathbf{X}&", + "\\mbox{(since $(f_S)_T=f_S$)}\\\\[4\\jot]", + "\\ar=\\dst\\int_Tf_S(\\mathbf{X})\\,d\\mathbf{X}&", + "\\mbox{(Definition~\\ref{thmtype:7.1.17}, applied to $f_S$ and $T$}),", + "\\end{array}", + "$$", + "which completes the proof." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", + "TRENCH_REAL_ANALYSIS-thmtype:7.1.17" + ], + "ref_ids": [ + 362, + 362, + 362 + ] + } + ], + "ref_ids": [] + }, + { + "id": 263, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.3", + "categories": [], + "title": "", + "contents": [ + "Suppose that $K$ is a bounded set with zero content and $\\epsilon,$", + "$\\rho>0.$ Then there are cubes $C_1,$ $C_2,$ \\dots$,$", + "$C_r$ with edge lengths", + "$<\\rho$ such that $C_j\\cap K\\ne\\emptyset,$ $1\\le j\\le r,$", + "\\begin{equation}\\label{eq:7.3.5}", + "K\\subset\\bigcup_{j=1}^r C_j,", + "\\end{equation}", + "and", + "$$", + "\\sum_{j=1}^r V(C_j)<\\epsilon.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $V(K)=0$,", + "$$", + "\\int_C\\psi_K(\\mathbf{X})\\,d\\mathbf{X}=0", + "$$", + "if $C$ is any cube containing $K$. From this and the", + "definition of the integral, there is a $\\delta>0$ such that if ${\\bf", + "P}$ is any partition of $C$ with $\\|{\\bf P}\\|\\le\\delta$ and $\\sigma$", + "is any Riemann sum of $\\psi_K$ over ${\\bf P}$, then", + "\\begin{equation}\\label{eq:7.3.6}", + "0\\le\\sigma\\le\\epsilon.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", + "into cubes with", + "\\begin{equation}\\label{eq:7.3.7}", + "\\|{\\bf P}\\|<\\min (\\rho,\\delta),", + "\\end{equation}", + "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\cap K\\ne", + "\\emptyset$ if $1\\le j\\le r$ and", + "$C_j\\cap K=\\emptyset$ if $r+1\\le j\\le k$. Then \\eqref{eq:7.3.5} holds, and", + "a typical Riemann sum of $\\psi_K$ over ${\\bf P}$ is of the form", + "$$", + "\\sigma=\\sum_{j=1}^r\\psi_K(\\mathbf{X}_j)V(C_j)", + "$$", + "with $\\mathbf{X}_j\\in C_j$, $1\\le j\\le r$. In particular, we", + "can choose", + "$\\mathbf{X}_j$ from $K$, so that $\\psi_K(\\mathbf{X}_j)=1$, and", + "$$", + "\\sigma=\\sum_{j=1}^r V(C_j).", + "$$", + "Now \\eqref{eq:7.3.6} and \\eqref{eq:7.3.7} imply that $C_1$, $C_2$, \\dots,", + "$C_r$ have the required properties." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 264, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.4", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}: \\R^n\\to \\R^n$ is continuously", + "differentiable on a bounded open set $S,$ and let $K$ be a closed", + "subset of $S$ with zero content$.$ Then $\\mathbf{G}(K)$ has zero content." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $K$ is a compact subset of the open set $S$, there is a", + " $\\rho_1>0$ such that the compact set", + "$$", + "K_{\\rho_1}=\\set{\\mathbf{X}}{\\dist(\\mathbf{X},K)\\le\\rho_1}", + "$$", + "is contained in $S$ (Exercise~5.1.26).", + "From", + "Lemma~\\ref{thmtype:6.2.7}, there is a constant $M$ such that", + "\\begin{equation}\\label{eq:7.3.8}", + "|\\mathbf{G}(\\mathbf{Y})-\\mathbf{G}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}|", + "\\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in K_{\\rho_1}.", + "\\end{equation}", + "Now suppose that $\\epsilon>0$. Since $V(K)=0$,", + "there are cubes $C_1$, $C_2$, \\dots, $C_r$ with edge", + "lengths", + "$s_1$, $s_2$, \\dots, $s_r<\\rho_1/\\sqrt n$ such that $C_j\\cap", + "K\\ne\\emptyset$, $1\\le j\\le r$,", + "$$", + "K\\subset\\bigcup_{j=1}^r C_j,", + "$$", + "and", + "\\begin{equation} \\label{eq:7.3.9}", + "\\sum_{j=1}^r V(C_j)<\\epsilon", + "\\end{equation}", + "(Lemma~\\ref{thmtype:7.3.3}). For $1\\le j\\le r$, let $\\mathbf{X}_j\\in C_j\\cap", + "K$. If $\\mathbf{X}\\in C_j$, then", + "$$", + "|\\mathbf{X}-\\mathbf{X}_j|\\le s_j\\sqrt n<\\rho_1,", + "$$", + "\\newpage", + "\\noindent", + "so $\\mathbf{X}\\in K$ and", + "$|\\mathbf{G}(\\mathbf{X})-\\mathbf{G}(\\mathbf{X}_j)|\\le M|\\mathbf{X}-\\mathbf{X}_j|\\le", + "M\\sqrt{n}\\,s_j$,", + "from \\eqref{eq:7.3.8}.", + "Therefore, $\\mathbf{G}(C_j)$ is contained in a cube", + "$\\widetilde{C}_j$ with edge length $2M\\sqrt{n}\\,s_j$,", + " centered at $\\mathbf{G}(\\mathbf{X}_j)$. Since", + "$$", + "V(\\widetilde{C}_j)=(2M\\sqrt{n})^ns_j^n=(2M\\sqrt{n})^nV(C_j),", + "$$", + "we now see that", + "$$", + "\\mathbf{G}(K)\\subset\\bigcup_{j=1}^r\\widetilde{C}_j", + "$$", + "and", + "$$", + "\\sum_{j=1}^r V(\\widetilde{C}_j)\\le", + "(2M\\sqrt{n})^n\\sum_{j=1}^r V(C_j)<(2M\\sqrt{n})^n\\epsilon,", + "$$", + "where the last inequality follows from \\eqref{eq:7.3.9}.", + "Since $(2M\\sqrt{n})^n$ does not depend on $\\epsilon$, it follows", + "that $V(\\mathbf{G}(K))=0$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.2.7", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.3" + ], + "ref_ids": [ + 258, + 263 + ] + } + ], + "ref_ids": [] + }, + { + "id": 265, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.6", + "categories": [], + "title": "", + "contents": [ + "A nonsingular $n\\times n$ matrix", + "$\\mathbf{A}$ can be written as", + "\\begin{equation}\\label{eq:7.3.10}", + "\\mathbf{A}=\\mathbf{E}_k\\mathbf{E}_{k-1}\\cdots\\mathbf{E}_1,", + "\\end{equation}", + "where each $\\mathbf{E}_i$ is a matrix that can be obtained from the", + "$n\\times n$ identity matrix $\\mathbf{I}$ by one of the following", + "operations$:$", + "\\begin{alist}", + "\\item % (a)", + "interchanging two rows of $\\mathbf{I};$", + "\\item % (b)", + "multiplying a row of $\\mathbf{I}$ by a nonzero constant$;$", + "\\item % (c)", + "adding a multiple of one row of $\\mathbf{I}$ to another$.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Theorem~\\ref{thmtype:7.3.5} implies that $\\mathbf{L}(S)$ is", + "Jordan measurable. If", + "\\begin{equation} \\label{eq:7.3.15}", + "V(\\mathbf{L}(R))=|\\det(\\mathbf{A})| V(R)", + "\\end{equation}", + "whenever $R$ is a rectangle, then", + " \\eqref{eq:7.3.14} holds if $S$", + "is any compact Jordan measurable set. To see this, suppose that", + "$\\epsilon>0$, let", + "$R$ be a rectangle containing $S$, and let", + "${\\bf P}=\\{R_1,R_2,\\dots,R_k\\}$ be a partition of $R$ such that the", + "upper and lower sums of $\\psi_S$ over ${\\bf", + "P}$ satisfy the inequality", + "\\begin{equation}\\label{eq:7.3.16}", + "S({\\bf P})-s({\\bf P})<\\epsilon.", + "\\end{equation}", + "Let ${\\mathcal U}_1$ and ${\\mathcal U}_2$ be as in \\eqref{eq:7.3.2}.", + "From \\eqref{eq:7.3.3} and \\eqref{eq:7.3.4},", + "\\begin{equation}\\label{eq:7.3.17}", + "s({\\bf P})=\\sum_{j\\in{\\mathcal U}_1} V(R_j)\\le V(S)\\le\\sum_{j\\in{\\mathcal U}_1} V(R_j)+\\sum_{j\\in{\\mathcal U}_2}", + "V(R_j)=S({\\bf P}).", + "\\end{equation}", + " Theorem~\\ref{thmtype:7.3.7}", + "implies that $\\mathbf{L}(R_1)$, $\\mathbf{L}(R_2)$, \\dots, $\\mathbf{L}(R_k)$", + "and", + "$\\mathbf{L}(S)$ are all Jordan measurable.", + "Since", + "$$", + "\\bigcup_{j\\in{\\mathcal U}_1}R_j\\subset S\\subset\\bigcup_{j\\in{\\mathcal", + "S}_1\\cup{\\mathcal S_2}}R_j,", + "$$", + "it follows that", + "$$", + "L\\left(\\bigcup_{j\\in{\\mathcal U}_1}R_j\\right)\\subset", + "L(S)\\subset L\\left(\\bigcup_{j\\in{\\mathcal S}_1\\cup{\\mathcal S_2}}R_j\\right).", + "$$", + "Since $L$ is one-to-one on $\\R^n$, this implies that", + "\\begin{equation} \\label{eq:7.3.18}", + "\\sum_{j\\in{\\mathcal U}_1} V(\\mathbf{L}(R_j))\\le V(\\mathbf{L}(S))\\le\\sum_{j\\in{\\mathcal U}_1}", + "V(\\mathbf{L}(R_j))+\\sum_{j\\in{\\mathcal U}_2} V(\\mathbf{L}(R_j)).", + "\\end{equation}", + "If we assume that \\eqref{eq:7.3.15} holds whenever $R$ is a rectangle,", + "then", + "$$", + "V(\\mathbf{L}(R_j))=|\\det(\\mathbf{A})|V(R_j),\\quad 1\\le j\\le k,", + "$$", + "so \\eqref{eq:7.3.18} implies that", + "$$", + "s({\\bf P})\\le \\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\le S({\\bf P}).", + "$$", + "This, \\eqref{eq:7.3.16} and \\eqref{eq:7.3.17} imply that", + "$$", + "\\left|V(S)-\\frac{V(\\mathbf{L}(S))}{ |\\det(\\mathbf{A})|}\\right|<\\epsilon;", + "$$", + "hence, since $\\epsilon$ can be made arbitrarily small, \\eqref{eq:7.3.14}", + "follows for any Jordan measurable set.", + "To complete the proof, we must verify \\eqref{eq:7.3.15} for every", + "rectangle", + "$$", + "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]=I_1\\times", + "I_2\\times\\cdots\\times I_n.", + "$$", + " Suppose that $\\mathbf{A}$ in \\eqref{eq:7.3.12} is an elementary matrix;", + "that is, let", + "$$", + "\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{EY}.", + "$$", + "{\\sc Case 1}. If $\\mathbf{E}$ is obtained by interchanging the $i$th and", + "$j$th rows of $\\mathbf{I}$, then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$ and $r\\ne j$};\\\\", + "y_j&\\mbox{if $r=i$};\\\\", + "y_i&\\mbox{if $r=j$}.\\end{array}\\right.", + "$$", + "Then $\\mathbf{L}(R)$ is the Cartesian product of $I_1$,", + "$I_2$, \\dots, $I_n$ with", + "$I_i$ and $I_j$ interchanged, so", + "$$", + "V(\\mathbf{L}(R))=V(R)=|\\det(\\mathbf{E})|V(R)", + "$$", + "since $\\det(\\mathbf{E})=-1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "{\\sc Case 2}. If $\\mathbf{E}$ is obtained by multiplying the $r$th row of", + "$\\mathbf{I}$ by $a$, then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$},\\\\", + "ay_i&\\mbox{if $r=i$}.\\end{array}\\right.", + "$$", + "Then", + "$$", + "\\mathbf{L}(R)=I_1\\times\\cdots\\times I_{i-1}\\times I'_i\\times I_{i+1}\\times", + "\\cdots\\times I_n,", + "$$", + "where $I'_i$ is an interval with length equal to $|a|$ times the", + "length of $I_i$, so", + "$$", + "V(\\mathbf{L}(R))=|a|V(R)=|\\det(\\mathbf{E})|V(R)", + "$$", + "since $\\det(\\mathbf{E})=a$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "{\\sc Case 3}. If $\\mathbf{E}$ is obtained by adding $a$ times the $j$th", + "row of $\\mathbf{I}$ to its $i$th row ($j\\ne i$), then", + "$$", + "x_r=\\left\\{\\casespace\\begin{array}{ll} y_r&\\mbox{if $r\\ne i$};\\\\", + "y_i+ay_j&\\mbox{if $r=i$}.\\end{array}\\right.", + "$$", + "Then", + "$$", + "\\mathbf{L}(R)=\\set{(x_1,x_2,\\dots,x_n)}{a_i+ax_j\\le x_i\\le b_i+ax_j", + "\\mbox{ and } a_r\\le x_r\\le b_r\\mbox{if } r\\ne i},", + "$$", + "which is a parallelogram if $n=2$ and a parallelepiped if $n=3$", + "(Figure~\\ref{figure:7.3.1}). Now", + "$$", + "V(\\mathbf{L}(R))=\\int_{\\mathbf{L}(R)} d\\mathbf{X},", + "$$", + "which we can evaluate as an iterated integral in which the first", + "integration is with respect to $x_i$. For example, if $i=1$, then", + "\\begin{equation}\\label{eq:7.3.19}", + "V(\\mathbf{L}(R))=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", + "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1+ax_j}_{a_1+ax_j} dx_1.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Since", + "$$", + "\\int^{b_1+ax_j}_{a_1+ax_j} dy_1=\\int^{b_1}_{a_1} dy_1,", + "$$", + "\\eqref{eq:7.3.19} can be rewritten as", + "\\begin{eqnarray*}", + "V(\\mathbf{L}(R))\\ar=\\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}}", + "dx_{n-1}\\cdots\\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} dx_1\\\\", + "\\ar=(b_n-a_n)(b_{n-1}-a_{n-1})\\cdots (b_1-a_1)=V(R).", + "\\end{eqnarray*}", + " Hence,", + "$V(\\mathbf{L}(R))=|\\det(\\mathbf{E})|V(R)$,", + "since $\\det(\\mathbf{E})=1$ in this case (Exercise~\\ref{exer:7.3.7}\\part{a}).", + "\\vskip12pt", + " \\centereps{3.6in}{4.6in}{fig070301.eps}", + " \\vskip6pt", + " \\refstepcounter{figure}", + " \\centerline{\\bf Figure \\thefigure} \\label{figure:7.3.1}", + " \\vskip12pt", + "From what we have shown so far, \\eqref{eq:7.3.14} holds if $\\mathbf{A}$ is an", + "elementary matrix and $S$ is any compact Jordan measurable set. If", + "$\\mathbf{A}$ is an arbitrary nonsingular matrix,", + "\\newpage", + "\\noindent", + "\\hskip -.0em", + "then we can write $\\mathbf{A}$", + "as a product of elementary matrices \\eqref{eq:7.3.10} and apply our known", + "result successively to $\\mathbf{L}_1$, $\\mathbf{L}_2$, \\dots, $\\mathbf{L}_k$", + "(see", + "\\eqref{eq:7.3.13}). This yields", + "$$", + "V(\\mathbf{L}(S))=|\\det(\\mathbf{E}_k)|\\,|\\det(\\mathbf{E}_{k-1})|\\cdots", + "|\\det\\mathbf{E}_1| V(S)=|\\det(\\mathbf{A})|V(S),", + "$$", + "by Theorem~\\ref{thmtype:6.1.9} and induction." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.3.5", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.7", + "TRENCH_REAL_ANALYSIS-thmtype:6.1.9" + ], + "ref_ids": [ + 215, + 216, + 173 + ] + } + ], + "ref_ids": [] + }, + { + "id": 266, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.10", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}:\\E^n\\to \\R^n$ is regular", + " on a cube $C$ in $\\E^n,$ and let $\\mathbf{A}$ be a", + "nonsingular $n\\times n$ matrix$.$ Then", + "\\begin{equation}\\label{eq:7.3.29}", + "V(\\mathbf{G}(C))\\le |\\det(\\mathbf{A})|\\left[\\max", + "\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", + "\\right]^n V(C).", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $s$ be the edge length of $C$. Let $\\mathbf{Y}_0=", + "(c_1,c_2,\\dots,c_n)$ be the center of $C$, and suppose that", + " $\\mathbf{H}=(y_1,y_2,\\dots,y_n)\\in C$.", + "If $\\mathbf{H}= (h_1,h_2,\\dots,h_n)$ is continuously differentiable on", + "$C$, then applying the mean value theorem", + "(Theorem~\\ref{thmtype:5.4.5}) to the components of", + "$\\mathbf{H}$ yields", + "$$", + "h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)=\\sum_{j=1}^n", + "\\frac{\\partial h_i(\\mathbf{Y}_i)}{\\partial y_j}(y_j-c_j),\\quad 1\\le i\\le n,", + "$$", + "where $\\mathbf{Y}_i\\in C$. Hence, recalling that", + "$$", + "\\mathbf{H}'(\\mathbf{Y})=\\left[\\frac{\\partial h_i}{\\partial", + "y_j}\\right]_{i,j=1}^n,", + "$$", + "applying Definition~\\ref{thmtype:7.3.9}, and noting that $|y_j-c_j|\\le", + "s/2$, $1\\le j\\le n$, we infer that", + "$$", + "|h_i(\\mathbf{Y})-h_i(\\mathbf{Y}_0)|\\le \\frac{s}{2}", + "\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C},\\quad 1\\le i\\le", + "n.", + "$$", + "This means that $\\mathbf{H}(C)$ is", + "contained in a cube with center $\\mathbf{X}_0=\\mathbf{H}(\\mathbf{Y}_0)$ and edge", + " length", + "$$", + "s\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}.", + "$$", + "Therefore,", + "\\begin{equation}\\label{eq:7.3.30}", + "\\begin{array}{rcl}", + "V(\\mathbf{H}(C))\\ar\\le", + "\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in", + "C} s^n\\\\[2\\jot]", + "\\ar=\\left[\\max\\set{\\|\\mathbf{H}'(\\mathbf{Y})\\|_\\infty\\right]^n}{\\mathbf{Y}\\in C}", + "V(C).", + "\\end{array}", + "\\end{equation}", + "Now let", + "$$", + "\\mathbf{L}(\\mathbf{X})=\\mathbf{A}^{-1}\\mathbf{X}", + "$$", + "and set $\\mathbf{H}=\\mathbf{L}\\circ\\mathbf{G}$; then", + "$$", + "\\mathbf{H}(C)=\\mathbf{L}(\\mathbf{G}(C))", + "\\mbox{\\quad and\\quad}\\mathbf{H}'=\\mathbf{A}^{-1}\\mathbf{G}',", + "$$", + "so \\eqref{eq:7.3.30} implies that", + "\\begin{equation}\\label{eq:7.3.31}", + "V(\\mathbf{L}(\\mathbf{G}(C)))\\le", + "\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C}", + "\\right]^nV(C).", + "\\end{equation}", + "Since $\\mathbf{L}$ is linear,", + "Theorem~\\ref{thmtype:7.3.7} with $\\mathbf{A}$ replaced by $\\mathbf{A}^{-1}$ implies that", + "$$", + "V(\\mathbf{L}(\\mathbf{G}(C)))=|\\det(\\mathbf{A})^{-1}|V(\\mathbf{G}(C)).", + "$$", + "This and \\eqref{eq:7.3.31} imply that", + "$$", + "|\\det(\\mathbf{A}^{-1})|V(\\mathbf{G}(C))", + "\\le\\left[\\max\\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in", + "C}", + "\\right]^nV(C).", + "$$", + "Since $\\det(\\mathbf{A}^{-1})=1/\\det(\\mathbf{A})$, this", + "implies \\eqref{eq:7.3.29}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.5", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", + "TRENCH_REAL_ANALYSIS-thmtype:7.3.7" + ], + "ref_ids": [ + 164, + 365, + 216 + ] + } + ], + "ref_ids": [] + }, + { + "id": 267, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.11", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{G}:\\E^n\\rightarrow \\R^n$", + " is regular on a cube $C$ in $\\R^n,$ then", + "\\begin{equation}\\label{eq:7.3.32}", + "V(\\mathbf{G}(C))\\le\\int_C |JG(\\mathbf{Y})|\\,d\\mathbf{Y}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let ${\\bf P}$ be a partition of $C$ into subcubes $C_1$, $C_2$,", + "\\dots, $C_k$ with centers $\\mathbf{Y}_1$, $\\mathbf{Y}_2$,", + "\\dots, $\\mathbf{Y}_k$. Then", + "\\begin{equation}\\label{eq:7.3.33}", + "V(\\mathbf{G}(C))=\\sum_{j=1}^k V(\\mathbf{G}(C_j)).", + "\\end{equation}", + "Applying Lemma~\\ref{thmtype:7.3.10}", + "to $C_j$ with $\\mathbf{A}=\\mathbf{G}'(\\mathbf{A}_j)$ yields", + "\\begin{equation}\\label{eq:7.3.34}", + "V(\\mathbf{G}(C_j))\\le |J\\mathbf{G}(\\mathbf{Y}_j)|", + "\\left[\\max\\set{\\|(\\mathbf{G}'(\\mathbf{Y}_j))^{-1}", + "\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C_j}", + "\\right]^n V(C_j).", + "\\end{equation}", + "Exercise~\\ref{exer:6.1.22} implies that if $\\epsilon>0$, there", + "is a $\\delta>0$ such that", + "$$", + "\\max\\set{\\|(\\mathbf{G}'(\\mathbf{Y}_j))^{-1}", + "\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C_j}", + "<1+\\epsilon,\\quad 1\\le j\\le k,\\mbox{\\quad if\\quad}\\|{\\bf P}\\|<\\delta.", + "$$", + "Therefore, from \\eqref{eq:7.3.34},", + "$$", + "V(\\mathbf{G}(C_j))\\le (1+\\epsilon)^n|J\\mathbf{G}(\\mathbf{Y}_j)|V(C_j),", + "$$", + " so \\eqref{eq:7.3.33} implies that", + "$$", + "V(\\mathbf{G}(C))\\le (1+\\epsilon)^n\\sum_{j=1}^k", + "|J\\mathbf{G}(\\mathbf{Y}_j)|V(C_j)\\mbox{\\quad if\\quad}\\|{\\bf P}\\|<\\delta.", + "$$", + "Since the sum on the right is a Riemann sum for", + " $\\int_C |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}$ and $\\epsilon$ can be", + "taken arbitrarily small, this implies \\eqref{eq:7.3.32}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.3.10" + ], + "ref_ids": [ + 266 + ] + } + ], + "ref_ids": [] + }, + { + "id": 268, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.12", + "categories": [], + "title": "", + "contents": [ + " Suppose that $S$ is Jordan measurable", + "and $\\epsilon,$ $\\rho>0.$ Then there are cubes", + "$C_1,$ $C_2,$ \\dots$,$ $C_r$ in $S$ with edge lengths $<\\rho,$ such", + "that $C_j\\subset S,$ $1\\le j\\le r,$", + "$C_i^0\\cap C_j^0=\\emptyset$ if $i\\ne j,$ and", + "\\begin{equation} \\label{eq:7.3.35}", + "V(S)\\le\\sum_{j=1}^r V(C_j)+\\epsilon.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $S$ is Jordan measurable,", + "$$", + "\\int_C\\psi_S(\\mathbf{X})\\,d\\mathbf{X}=V(S)", + "$$", + "if $C$ is any cube containing $S$. From this and the", + "definition of the integral, there is a $\\delta>0$ such that if ${\\bf", + "P}$ is any partition of $C$ with $\\|{\\bf P}\\|<\\delta$ and $\\sigma$", + "is any Riemann sum of $\\psi_S$ over ${\\bf P}$, then", + "$\\sigma>V(S)-\\epsilon/2$. Therefore, if $s(P)$ is the lower sum of", + "$\\psi_S$ over $\\mathbf{P}$, then", + "\\begin{equation} \\label{eq:7.3.36}", + "s(\\mathbf{P})>V(S)-\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{P}\\|<\\delta.", + "\\end{equation}", + "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", + "into cubes with", + "$\\|{\\bf P}\\|<\\min (\\rho,\\delta)$,", + "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\subset", + "S$ if", + " $1\\le j\\le r$ and $C_j\\cap S^c\\ne\\emptyset$ if $j>r$.", + "From \\eqref{eq:7.3.4}, $s(\\mathbf{P})=\\sum_{j=1}^rV(C_k)$. This and", + "\\eqref{eq:7.3.36} imply \\eqref{eq:7.3.35}. Clearly, $C_i^0\\cap", + "C_j^0=\\emptyset$ if $i\\ne j$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 269, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.13", + "categories": [], + "title": "", + "contents": [ + "Suppose that $\\mathbf{G}: \\E^n\\to \\R^n$ is regular on a", + "compact Jordan measurable set $S$ and $f$ is continuous and", + "nonnegative on", + "$\\mathbf{G}(S).$", + "Let", + "\\begin{equation}\\label{eq:7.3.37}", + "Q(S)=\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_S", + " f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", + "\\end{equation}", + "Then $Q(S)\\le0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From the continuity of $J\\mathbf{G}$ and $f$ on the compact sets $S$ and", + "$\\mathbf{G}(S)$, there are constants $M_1$ and $M_2$ such that", + "\\begin{equation}\\label{eq:7.3.38}", + "|J\\mathbf{G}(\\mathbf{Y})|\\le M_1\\mbox{\\quad if\\quad}\\mathbf{Y}\\in S", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.3.39}", + "|f(\\mathbf{X})|\\le M_2\\mbox{\\quad if\\quad}\\mathbf{X}\\in\\mathbf{G}(S)", + "\\end{equation}", + " (Theorem~\\ref{thmtype:5.2.11}).", + "Now suppose that $\\epsilon>0$. Since", + "$f\\circ\\mathbf{G}$ is uniformly continuous on $S$", + "(Theorem~\\ref{thmtype:5.2.14}),", + " there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:7.3.40}", + "|f(\\mathbf{G}(\\mathbf{Y}))-f(\\mathbf{G}(\\mathbf{Y}'))|<\\epsilon", + "\\mbox{\\quad if \\quad$|\\mathbf{Y}-\\mathbf{Y}'|<\\delta$", + "and }\\mathbf{Y},\\mathbf{Y}' \\in S.", + "\\end{equation}", + "Now let $C_1$, $C_2$, \\dots, $C_r$ be chosen as described in", + "Lemma~\\ref{thmtype:7.3.12}, with $\\rho=\\delta/\\sqrt{n}$.", + " Let", + "$$", + "S_1=\\set{\\mathbf{Y}\\in S}{\\mathbf{Y}\\notin\\bigcup_{j=1}^r C_j}.", + "$$", + "Then $V(S_1)<\\epsilon$ and", + "\\begin{equation} \\label{eq:7.3.41}", + "S=\\left(\\bigcup_{j=1}^r C_j\\right)\\cup S_1.", + "\\end{equation}", + "Suppose that $\\mathbf{Y}_1$, $\\mathbf{Y}_2$, \\dots, $\\mathbf{Y}_r$ are points in", + "$C_1$, $C_2$, \\dots, $C_r$ and $\\mathbf{X}_j=\\mathbf{G}(\\mathbf{Y}_j)$, $1\\le", + "j\\le r$. From", + "\\eqref{eq:7.3.41} and Theorem~\\ref{thmtype:7.1.30},", + "\\begin{eqnarray*}", + "Q(S)\\ar=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_{S_1}", + "f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y} \\\\", + "\\ar{}+\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)} f(\\mathbf{X})\\,d\\mathbf{X}-", + "\\sum_{j=1}^r\\int_{C_j}", + "f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", + "\\ar=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_{S_1}", + " f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", + "\\ar{}+\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)}(f(\\mathbf{X})-", + "f(\\mathbf{A}_j))\\,d\\mathbf{X}\\\\", + "\\ar{}+\\sum_{j=1}^r\\int_{C_j}((f(\\mathbf{G}(\\mathbf{Y}_j))-", + "f(\\mathbf{G}(\\mathbf{Y})))|J(\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\\\", + "\\ar{}+\\sum_{j=1}^r f(\\mathbf{X}_j)\\left(V(\\mathbf{G}(C_j))-", + "\\int_{C_j} |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\right).", + "\\end{eqnarray*}", + "\\newpage", + "\\noindent", + "Since $f(\\mathbf{X})\\ge0$,", + "$$", + "\\int_{S_1}f(\\mathbf{G}(\\mathbf{Y}))|J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}\\ge0,", + "$$", + "and", + "Lemma~\\ref{thmtype:7.3.11}", + "implies that the last", + "sum is nonpositive.", + "Therefore,", + "\\begin{equation} \\label{eq:7.3.42}", + "Q(S)\\le I_1+I_2+I_3,", + "\\end{equation}", + "where", + "$$", + "I_1=\\int_{\\mathbf{G}(S_1)} f(\\mathbf{X})\\,d\\mathbf{X},\\quad", + "I_2=", + "\\sum_{j=1}^r\\int_{\\mathbf{G}(C_j)}|f(\\mathbf{X})-f(\\mathbf{X}_j)|", + "\\,d\\mathbf{X},", + "$$", + "and", + "$$", + "I_3=", + "\\sum_{j=1}^r\\int_{C_j}|f(\\mathbf{G})(\\mathbf{Y}_j))-f(\\mathbf{G}(\\mathbf{Y}))|", + " |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}.", + "$$", + "We will now estimate these three terms. Suppose that $\\epsilon>0$.", + "To estimate $I_1$, we first remind you that since $\\mathbf{G}$", + "is regular on the compact set $S$, $\\mathbf{G}$ is also regular on some", + "open", + "set ${\\mathcal O}$ containing $S$ (Definition~\\ref{thmtype:6.3.2}).", + "Therefore, since $S_1\\subset S$ and $V(S_1)<\\epsilon$,", + "$S_1$ can be covered by cubes $T_1$, $T_2$, \\dots, $T_m$ such that", + "\\begin{equation} \\label{eq:7.3.43}", + "\\sum_{j=1}^r V(T_j)< \\epsilon", + "\\end{equation}", + " and $\\mathbf{G}$ is regular on $\\bigcup_{j=1}^m", + "T_j$. Now,", + "$$", + "\\begin{array}{rcll}", + "I_1\\ar\\le M_2V(\\mathbf{G}(S_1))& \\mbox{(from", + "\\eqref{eq:7.3.39})}\\\\[2\\jot]", + "\\ar\\le M_2\\dst\\sum_{j=1}^m V(\\mathbf{G}(T_j))&(\\mbox{since", + "}S_1\\subset\\cup_{j=1}^mT_j)\\\\[2\\jot]", + "\\ar\\le M_2\\dst\\sum_{j=1}^m\\int_{T_j}| J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}&", + "\\mbox{(from Lemma~\\ref{thmtype:7.3.11})}", + "\\\\[2\\jot]", + "\\ar\\le M_2M_1\\epsilon& \\mbox{(from \\eqref{eq:7.3.38}", + "and", + "\\eqref{eq:7.3.43})}.", + "\\end{array}", + "$$", + "To estimate $I_2$, we note that", + "if $\\mathbf{X}$ and $\\mathbf{X}_j$ are in $\\mathbf{G}(C_j)$", + "then $\\mathbf{X}=\\mathbf{G}(\\mathbf{Y})$ and", + "$\\mathbf{X}_j=\\mathbf{G}(\\mathbf{Y}_j)$ for some $\\mathbf{Y}$ and $\\mathbf{Y}_j$ in", + "$C_j$. Since the edge length of $C_j$ is less than", + "$\\delta/\\sqrt n$, it follows that $|\\mathbf{Y}-\\mathbf{Y}_j|<\\delta$, so", + " $|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\epsilon$, by \\eqref{eq:7.3.40}.", + "Therefore,", + "$$", + "\\begin{array}{rcll}", + "I_2\\ar< \\epsilon\\dst\\sum_{j=1}^r V(\\mathbf{G}(C_j))\\\\[2\\jot]", + "\\ar\\le \\epsilon\\dst\\sum_{j=1}^r\\int_{C_j}|J\\mathbf{G}(\\mathbf{Y})|d\\mathbf{Y}&", + "\\mbox{(from Lemma~\\ref{thmtype:7.3.11})}\\\\[2\\jot]", + "\\ar\\le \\dst\\epsilon M_1\\sum_{j=1}^r V(C_j)&\\mbox{(from", + "\\eqref{eq:7.3.38}})\\\\[2\\jot]", + "\\ar\\le \\epsilon M_1 V(S)&(\\mbox{since }\\dst\\cup_{j=1}^rC_j\\subset S).", + "\\end{array}", + "$$", + "\\newpage", + "To estimate $I_3$, we note again from \\eqref{eq:7.3.40} that", + " $|f(\\mathbf{G}(\\mathbf{Y}_j))-f(\\mathbf{G}(\\mathbf{Y}))|<", + " \\epsilon$ if $\\mathbf{Y}$ and $\\mathbf{Y}_j$ are in $C_j$.", + " Hence,", + "\\begin{eqnarray*}", + "I_3\\ar< \\epsilon\\sum_{j=1}^r", + "\\int_{C_j}|J\\mathbf{G}(\\mathbf{Y})|d\\mathbf{Y}\\\\", + "\\ar\\le M_1\\epsilon\\sum_{j=1}^r V(C_j)", + "\\mbox{\\quad(from \\eqref{eq:7.3.38}}\\\\", + "\\ar\\le M_1 V(S)\\epsilon", + "\\end{eqnarray*}", + "because $\\bigcup_{j=1}^r C_j\\subset S$ and $C_i^0\\cap C_j^0=\\emptyset$", + "if", + "$i\\ne j$.", + "From these inequalities on $I_1$, $I_2$, and $I_3$,", + "\\eqref{eq:7.3.42} now implies that", + "$$", + "Q(S)1$ and", + "$q=p/(p-1);$ thus$,$", + "\\begin{equation} \\label{eq:8.1.5}", + "\\frac{1}{p}+\\frac{1}{q}=1.", + "\\end{equation}", + " Then", + "\\begin{equation} \\label{eq:8.1.6}", + "\\sum_{i=1}^n \\mu_i\\nu_i\\le\\left(\\sum_{i=1}^n\\mu_i^p\\right)^{1/p}", + "\\left(\\sum_{i=1}^n \\nu_i^q\\right)^{1/q}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $\\alpha$ and $\\beta$ be any two positive numbers, and", + "consider the function", + "$$", + "f(\\beta)=\\frac{\\alpha^p}{p}+\\frac{\\beta^q}{q}-\\alpha\\beta,", + "$$", + "\\newpage", + "\\noindent", + "where we regard $\\alpha$ as a constant. Since $f'(\\beta)=\\beta^{q-1}-\\alpha$ and", + "$f''(\\beta)=(q-1)\\beta^{q-2}>0$ for $\\beta>0$, $f$ assumes its minimum value", + "on $[0,\\infty)$ at $\\beta=\\alpha^{1/(q-1)}=\\alpha^{p-1}$. But", + "$$", + "f(\\alpha^{p-1})=\\frac{\\alpha^p}{p}+\\frac{\\alpha^{(p-1)q}}{q}-\\alpha^p", + "=\\alpha^p\\left(\\frac{1}{p}+\\frac{1}{q}-1\\right)=0.", + "$$", + "Therefore,", + "\\begin{equation} \\label{eq:8.1.7}", + "\\alpha\\beta\\le \\frac{\\alpha^p}{p}+\\frac{\\beta^q}{q}\\mbox{\\quad if \\quad}", + "\\alpha, \\beta\\ge0.", + "\\end{equation}", + "Now let", + "$$", + "\\alpha_i=\\mu_i\\left(\\sum_{j=1}^n \\mu_j^p\\right)^{-1/p}", + "\\mbox{\\quad and \\quad}", + "\\beta_i=\\nu_i\\left(\\sum_{j=1}^n \\nu_j^q\\right)^{-1/q}.", + "$$", + "From \\eqref{eq:8.1.7},", + "$$", + "\\alpha_i\\beta_i\\le\\frac{\\mu_i^p}{p}\\left(\\sum_{j=1}^n \\mu_j^p\\right)^{-1}", + "+\\frac{\\nu_i^q}{q}\\left(\\sum_{j=1}^n \\nu_j^q\\right)^{-1}.", + "$$", + "From \\eqref{eq:8.1.5}, summing this from $i=1$ to $n$ yields $\\sum_{i=1}^n", + "\\alpha_i\\beta_i\\le1$, which implies", + "\\eqref{eq:8.1.6}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 272, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.8", + "categories": [], + "title": "", + "contents": [ + "Suppose that $u_1,$ $u_2,$ \\dots$,$ $u_n$ and $v_1,$ $v_2,$ \\dots$,$ $v_n$", + "are nonnegative numbers and $p>1.$ Then", + "\\begin{equation} \\label{eq:8.1.8}", + "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/p}", + "\\le\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", + "+\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Again, let $q=p/(p-1)$. We write", + "\\begin{equation} \\label{eq:8.1.9}", + "\\sum_{i=1}^n(u_i+v_i)^p=\\sum_{i=1}^n u_i(u_i+v_i)^{p-1}", + "+\\sum_{i=1}^n v_i(u_i+v_i)^{p-1}.", + "\\end{equation}", + "From H\\\"older's inequality with $\\mu_i=u_i$ and", + "$\\nu_i=(u_i+v_i)^{p-1}$,", + "\\begin{equation} \\label{eq:8.1.10}", + "\\sum_{i=1}^n u_i(u_i+v_i)^{p-1}\\le", + "\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", + "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q},", + "\\end{equation}", + "since $q(p-1)=p$. Similarly,", + "$$", + "\\sum_{i=1}^n v_i(u_i+v_i)^{p-1}\\le", + "\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}", + "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q}.", + "$$", + "This, \\eqref{eq:8.1.9}, and \\eqref{eq:8.1.10} imply that", + "$$", + "\\sum_{i=1}^n(u_i+v_i)^p", + "\\le\\left[\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p}", + "+\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}\\right]", + "\\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/q}.", + "$$", + "\\newpage", + "\\noindent", + "Since $1-1/q=1/p$, this implies \\eqref{eq:8.1.8}, which is", + "known as {\\it Minkowski's inequality\\/}." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 273, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.2", + "categories": [], + "title": "", + "contents": [ + "If $a$ and $b$ are any two real numbers$,$ then", + "\\begin{equation} \\label{eq:1.1.4}", + "|a-b|\\ge\\big||a|-|b|\\big|", + "\\end{equation}", + "and", + "\\begin{equation} \\label{eq:1.1.5}", + "|a+b|\\ge\\big||a|-|b|\\big|.", + "\\end{equation}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Replacing $a$ by $a-b$ in \\eqref{eq:1.1.3} yields", + "$$", + "|a|\\le|a-b|+|b|,", + "$$", + "so", + "\\begin{equation} \\label{eq:1.1.6}", + "|a-b|\\ge|a|-|b|.", + "\\end{equation}", + "Interchanging $a$ and $b$ here yields", + "$$", + "|b-a|\\ge|b|-|a|,", + "$$", + "which is equivalent to", + "\\begin{equation} \\label{eq:1.1.7}", + "|a-b|\\ge|b|-|a|,", + "\\end{equation}", + "since $|b-a|=|a-b|$. Since", + "$$", + "\\big||a|-|b|\\big|=", + "\\left\\{\\casespace\\begin{array}{l} |a|-|b|\\mbox{\\quad if \\quad} |a|>|b|,\\\\[2\\jot]", + " |b|-|a|\\mbox{\\quad if \\quad} |b|>|a|,", + "\\end{array}\\right.", + "$$", + "\\eqref{eq:1.1.6} and \\eqref{eq:1.1.7} imply \\eqref{eq:1.1.4}. Replacing", + "$b$ by $-b$ in \\eqref{eq:1.1.4} yields \\eqref{eq:1.1.5}, since", + "$|-b|=|b|$." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 274, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.6", + "categories": [], + "title": "", + "contents": [ + "contains all its limit points$.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $S$ is bounded, it has an infimum $\\alpha$", + "and a supremum $\\beta$, and, since $S$ is closed, $\\alpha$", + "and $\\beta$ belong to $S$ (Exercise~\\ref{exer:1.3.17}). Define", + "$$", + "S_t=S\\cap [\\alpha,t] \\mbox{\\quad for \\ } t\\ge\\alpha,", + "$$", + "and let", + "$$", + "F=\\set{t}{\\alpha\\le t\\le\\beta \\mbox{\\ and finitely many sets from", + "${\\mathcal H}$ cover $S_t$}}.", + "$$", + "Since $S_\\beta=S$, the theorem will be proved if we can show that", + "$\\beta", + "\\in F$. To do this, we use the completeness of the reals.", + "Since $\\alpha\\in S$, $S_\\alpha$ is the singleton set $\\{\\alpha\\}$,", + "which is contained in some open set $H_\\alpha$ from ${\\mathcal H}$", + "because ${\\mathcal H}$ covers $S$; therefore, $\\alpha\\in F$. Since $F$ is", + "nonempty and bounded above by $\\beta$, it has a supremum $\\gamma$.", + "First, we wish to show that $\\gamma=\\beta$. Since $\\gamma\\le\\beta$ by", + "definition of $F$, it suffices to rule out the possibility that", + "$\\gamma<\\beta$. We consider two cases.", + "{\\sc Case 1}. Suppose that $\\gamma<\\beta$ and $\\gamma\\not\\in S$. Then,", + "since $S$ is closed, $\\gamma$ is not a limit point of $S$", + "(Theorem~\\ref{thmtype:1.3.5}). Consequently, there is an $\\epsilon>0$", + "such that", + "$$", + "[\\gamma-\\epsilon,\\gamma+\\epsilon]\\cap S=\\emptyset,", + "$$", + "so $S_{\\gamma-\\epsilon}=S_{\\gamma+\\epsilon}$. However, the", + "definition of $\\gamma$ implies that $S_{\\gamma-\\epsilon}$ has a finite", + "subcovering from ${\\mathcal H}$, while $S_{\\gamma+\\epsilon}$ does not.", + "This is a contradiction.", + "{\\sc Case 2}. Suppose that $\\gamma<\\beta$ and $\\gamma\\in S$. Then", + "there is an open", + "set $H_\\gamma$ in ${\\mathcal H}$ that contains $\\gamma$ and, along with $\\gamma$, an", + "interval $[\\gamma-\\epsilon,\\gamma+\\epsilon]$ for some positive", + "$\\epsilon$.", + "Since $S_{\\gamma-\\epsilon}$ has a finite covering $\\{H_1, \\dots,H_n\\}$ of", + "sets from ${\\mathcal H}$, it follows that $S_{\\gamma+\\epsilon}$ has the finite", + "covering $\\{H_1, \\dots,H_n,H_\\gamma\\}$. This contradicts the", + "definition of $\\gamma$.", + "Now we know that $\\gamma=\\beta$, which is in $S$. Therefore, there is", + "an open set $H_\\beta$ in ${\\mathcal H}$ that contains $\\beta$ and along", + "with $\\beta$, an interval of the form", + "$[\\beta-\\epsilon,\\beta+\\epsilon]$, for some positive $\\epsilon$. Since", + "$S_{\\beta-\\epsilon}$ is covered by a finite collection of sets", + "$\\{H_1, \\dots,H_k\\}$, $S_\\beta$ is covered by the finite collection", + "$\\{H_1, \\dots, H_k, H_\\beta\\}$. Since $S_\\beta=S$, we are", + "finished." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:1.3.5" + ], + "ref_ids": [ + 10 + ] + } + ], + "ref_ids": [] + }, + { + "id": 275, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.2.13", + "categories": [], + "title": "", + "contents": [ + "If $f$ is continuous on a set $T,$ then $f$ is uniformly continuous", + "on any finite closed interval contained in $T.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We assume that $f$ is nondecreasing, and", + "leave the case where $f$ is nonincreasing to you", + "(Exercise~\\ref{exer:2.2.34}).", + "Theorem~\\ref{thmtype:2.1.9}\\part{a}", + "implies that the set $\\widetilde R_f=\\set{f(x)}{x\\in(a,b)}$", + "is a subset of the open interval $(f(a+),f(b-))$. Therefore,", + "\\begin{equation} \\label{eq:2.2.16}", + "R_f=\\{f(a)\\}\\cup\\widetilde", + "R_f\\cup\\{f(b)\\}\\subset\\{f(a)\\}\\cup(f(a+),f(b-))\\cup\\{f(b)\\}.", + "\\end{equation}", + "Now", + "suppose that $f$ is continuous on $[a,b]$. Then $f(a)=f(a+)$,", + "$f(b-)=f(b)$,", + "so \\eqref{eq:2.2.16} implies that", + "$R_f\\subset[f(a),f(b)]$. If $f(a)<\\mu0$ there is an integer", + "$K$ such that", + "$$", + "\\left|\\sum_{n=k}^\\infty a_n\\right|<\\epsilon\\mbox{\\quad if\\quad} k\\ge", + "K;", + "$$", + "that is$,$", + "$$", + "\\lim_{k\\to\\infty}\\sum_{n=k}^\\infty a_n=0.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since $A_n=A_{n-1}+a_n$ and $a_n\\ge0$ $(n\\ge k)$, the sequence", + "$\\{A_n\\}$ is nondecreasing, so the conclusion follows from", + "Theorem~\\ref{thmtype:4.1.6}\\part{a} and", + "Definition~\\ref{thmtype:4.3.1}.", + "\\newline\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.1.6", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.1" + ], + "ref_ids": [ + 83, + 329 + ] + } + ], + "ref_ids": [] + }, + { + "id": 279, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.12", + "categories": [], + "title": "", + "contents": [ + "Suppose that $a_n\\ge0$ and $b_n>0$ for $n\\ge k,$ and", + "$$", + "\\lim_{n\\to\\infty}\\frac{a_n}{ b_n}=L,", + "$$", + "where $00\\ (n\\ge k)$ and", + "$$", + "\\lim_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}=L.", + "$$", + "\\vskip-1em", + "Then", + "\\begin{alist}", + "\\item % (a)", + " $\\sum a_n<\\infty$ if $L<1.$", + "\\item % (b)", + " $\\sum a_n=\\infty$ if $L>1.$", + "\\end{alist}", + "The test is inconclusive if $L=1.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "\\part{a}", + "We need the inequality", + "\\begin{equation}\\label{eq:4.3.15}", + "\\frac{1}{(1+x)^p}>1-px,\\quad x>0,\\ p>0.", + "\\end{equation}", + "This follows from Taylor's theorem", + "(Theorem~\\ref{thmtype:2.5.4}), which implies that", + "$$", + "\\frac{1}{(1+x)^p}=1-px+\\frac{1}{2}\\frac{p(p+1)}{(1+c)^{p+2}}x^2,", + "$$", + "where $00$,", + "this implies \\eqref{eq:4.3.15}.", + "Now suppose that $M<-p<-1$. Then there is an integer $k$ such that", + "$$", + "n\\left(\\frac{a_{n+1}}{ a_n}-1\\right)<-p,\\quad n\\ge k,", + "$$", + "so", + "$$", + "\\frac{a_{n+1}}{ a_n}<1-\\frac{p}{ n},\\quad n\\ge k.", + "$$", + "Hence,", + "$$", + "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(1+1/n)^p},\\quad n\\ge k,", + "$$", + "as can be seen by letting $x=1/n$ in \\eqref{eq:4.3.15}. From this,", + "$$", + "\\frac{a_{n+1}}{ a_n}<\\frac{1}{(n+1)^p}\\bigg/\\frac{1}{ n^p},\\quad n\\ge k.", + "$$", + " Since $\\sum 1/n^p<\\infty$ if $p>1$,", + " Theorem~\\ref{thmtype:4.3.13}\\part{a} implies that", + " $\\sum a_n<\\infty$.", + "\\part{b} Here we need the inequality", + "\\begin{equation}\\label{eq:4.3.16}", + "(1-x)^q<1-qx,\\quad 0-q,\\quad n\\ge k,", + "$$", + "so", + "$$", + "\\frac{a_{n+1}}{ a_n}\\ge1-\\frac{q}{ n},\\quad n\\ge k.", + "$$", + "If $q\\le0$, then $\\sum a_n=\\infty$, by Corollary~\\ref{thmtype:4.3.6}.", + "Hence, we may assume that $0\\left(1-\\frac{1}{ n}\\right)^q,\\quad n\\ge k,", + "$$", + "\\newpage", + "\\noindent", + "as can be seen by setting $x=1/n$ in \\eqref{eq:4.3.16}. Hence,", + "$$", + "\\frac{a_{n+1}}{ a_n}>\\frac{1}{ n^q}\\bigg/\\frac{1}{(n-1)^q},\\quad n\\ge k.", + "$$", + " Since $\\sum 1/n^q=\\infty$ if $q<1$,", + " Theorem~\\ref{thmtype:4.3.13}\\part{b} implies that", + " $\\sum a_n=\\infty$." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:2.5.4", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.6", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.13" + ], + "ref_ids": [ + 42, + 103, + 277, + 103 + ] + } + ], + "ref_ids": [] + }, + { + "id": 281, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.21", + "categories": [], + "title": "", + "contents": [ + "The series $\\sum a_nb_n$ converges if $a_{n+1}\\le a_n$ for $n\\ge k,$", + "$\\lim_{n\\to\\infty}a_n=0,$ and", + "$$", + "|b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k,", + "$$", + "for some constant $M.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "If $a_{n+1}\\le a_n$, then", + "$$", + "\\sum_{n=k}^m |a_{n+1}-a_n|=\\sum_{n=k}^m (a_n-a_{n+1})=a_k-a_{m+1}.", + "$$", + "Since $\\lim_{m\\to\\infty} a_{m+1}=0$, it follows that", + "$$", + "\\sum_{n=k}^\\infty |a_{n+1}-a_n|=a_k<\\infty.", + "$$", + "Therefore, the hypotheses of Dirichlet's test are satisfied,", + "so $\\sum a_nb_n$ converges." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 282, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.3.22", + "categories": [], + "title": "Alternating Series Test", + "contents": [ + "The series $\\sum (-1)^na_n$ converges if $0\\le a_{n+1}\\le a_n$ and", + "$\\lim_{n\\to\\infty} a_n=0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let $b_n=(-1)^n$; then $\\{|B_n|\\}$ is a sequence of zeros and", + "ones and therefore bounded. The conclusion now follows from", + "Abel's test." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 283, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.8", + "categories": [], + "title": "", + "contents": [ + "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is", + "continuous on $S,$ then so is $F;$ that is$,$ a uniform limit of", + "continuous functions is continuous." + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Since", + "\\begin{eqnarray*}", + "\\left|\\int_a^b F_n(x)\\,dx-\\int_a^b F(x)\\,dx\\right|\\ar\\le \\int_a^b", + "|F_n(x)-F(x)|\\,dx\\\\", + "\\ar\\le (b-a)\\|F_n-F\\|_S", + "\\end{eqnarray*}", + "and $\\lim_{n\\to\\infty}\\|F_n-F\\|_S=0$, the conclusion follows." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + }, + { + "id": 284, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.14", + "categories": [], + "title": "", + "contents": [ + "If $\\sum f_n$ converges uniformly on $S,$ then", + "$\\lim_{n\\to\\infty}\\|f_n\\|_S=0.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "From Cauchy's convergence criterion for series of constants,", + "there is for each $\\epsilon>0$ an integer $N$ such that", + "$$", + "M_n+M_{n+1}+\\cdots+M_m<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N,", + "$$", + "which, because of \\eqref{eq:4.4.17}, implies that", + "$$", + "\\|f_n\\|_S+\\|f_{n+1}\\|_S+\\cdots+\\|f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad}", + " m, n\\ge N.", + "$$", + " Lemma~\\ref{thmtype:4.4.2} and Theorem~\\ref{thmtype:4.4.13} imply that", + "$\\sum f_n$ converges uniformly on $S$.", + "\\mbox{}" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.2", + "TRENCH_REAL_ANALYSIS-thmtype:4.4.13" + ], + "ref_ids": [ + 251, + 122 + ] + } + ], + "ref_ids": [] + }, + { + "id": 285, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.17", + "categories": [], + "title": "", + "contents": [ + "The series $\\sum_{n=k}^\\infty f_ng_n$ converges uniformly on $S$ if", + "$$", + "f_{n+1}(x)\\le f_n(x),\\quad x\\in S,\\quad n\\ge k,", + "$$", + "$\\{f_n\\}$ converges uniformly to zero on $S,$ and", + "$$", + "\\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k,", + "$$", + "for some constant $M.$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "In any case, the series \\eqref{eq:4.5.1} converges to $a_0$ if", + "$x=x_0$. If", + "\\begin{equation}\\label{eq:4.5.3}", + "\\sum |a_n|r^n<\\infty", + "\\end{equation}", + "for some $r>0$, then $\\sum a_n (x-x_0)^n$ converges", + "absolutely uniformly in $[x_0-r, x_0+r]$, by Weierstrass's test", + "(Theorem~\\ref{thmtype:4.4.15}) and", + "Exercise~\\ref{exer:4.4.21}. From Cauchy's root test", + "(Theorem~\\ref{thmtype:4.3.17}),", + "\\eqref{eq:4.5.3} holds if", + "$$", + "\\limsup_{n\\to\\infty} (|a_n|r^n)^{1/n}<1,", + "$$", + "which is equivalent to", + " $$", + " r\\,\\limsup_{n\\to\\infty} |a_n|^{1/n}<1", + "$$", + "(Exercise~\\ref{exer:4.1.30}\\part{a}).", + "From \\eqref{eq:4.5.2}, this can be rewritten as $rR$, then", + "\\newpage", + "$$", + "\\frac{1}{ R}>\\frac{1}{ |x-x_0|},", + "$$", + "so \\eqref{eq:4.5.2} implies that", + "$$", + "|a_n|^{1/n}\\ge\\frac{1}{ |x-x_0|}\\mbox{\\quad and therefore\\quad}", + "|a_n(x-x_0)^n|\\ge1", + "$$", + "for infinitely many values of $n$. Therefore, $\\sum a_n(x-x_0)^n$", + "diverges (Corollary~\\ref{thmtype:4.3.6}) if $|x-x_0|>R$.", + "In particular, the series diverges for all $x\\ne x_0$ if $R=0$.", + "To prove the assertions concerning the possibilities at $x=x_0+R$", + "and $x=x_0-R$ requires examples, which follow. (Also, see", + "Exercise~\\ref{exer:4.5.1}.)" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:4.4.15", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.17", + "TRENCH_REAL_ANALYSIS-thmtype:4.3.6" + ], + "ref_ids": [ + 123, + 106, + 277 + ] + } + ], + "ref_ids": [] + }, + { + "id": 286, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.6", + "categories": [], + "title": "", + "contents": [ + "If", + "$$", + "f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n,\\quad |x-x_0|0;$ $(x_0,y_0)$", + "is a local minimum point if $f_{xx}(x_0,y_0)>0$, or a local maximum", + "point if", + "$f_{xx}(x_0,y_0)<0.$", + "\\item % (b)", + "$(x_0,y_0)$ is not a local extreme point of $f$ if $D<0.$", + "\\end{alist}" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Write $(x-x_0,y-y_0)=(u,v)$ and", + "$$", + "p(u,v)=(d^{(2)}_{\\mathbf{X}_0}f)(u,v)=Au^2+2Buv+Cv^2,", + "$$", + "where $A=f_{xx}(x_0,y_0)$, $B=f_{xy}(x_0,y_0)$, and", + "$C=f_{yy}(x_0,y_0)$, so", + "$$", + "D=AC-B^2.", + "$$", + "If $D>0$, then $A\\ne0$, and we can write", + "\\begin{eqnarray*}", + "p(u,v)\\ar=A\\left(u^2+\\frac{2B}{ A} uv+\\frac{B^2}{", + "A^2}v^2\\right)+\\left(C-\\frac{B^2}{ A}\\right)v^2\\\\", + "\\ar=A\\left(u+\\frac{B}{ A}v\\right)^2+\\frac{D}{ A}v^2.", + "\\end{eqnarray*}", + "This cannot vanish unless $u=v=0$. Hence, $d^{(2)}_{\\mathbf{X}_0}f$ is", + "positive definite if $A>0$ or negative definite if $A<0$, and", + "Theorem~\\ref{thmtype:5.4.10}\\part{b} implies \\part{a}.", + "If $D<0$, there are three possibilities:", + "\\newpage", + "\\begin{description}", + "\\item{\\bf 1.} $A\\ne0$; then $p(1,0)=A$ and", + "$\\dst{p\\left(-\\frac{B}{ A},1\\right)=\\frac{D}{ A}}$.", + "\\vspace*{6pt}", + "\\item{\\bf 2.} $C\\ne0$; then $p(0,1)=C$ and $\\dst{p\\left(1,", + "-\\frac{B}{ C}\\right)=\\frac{D}{ C}}$.", + "\\vspace*{6pt}", + "\\item{\\bf 3.} $A=C=0$; then $B\\ne0$ and $p(1,1)=2B$ and $p(1,-1)=-2B$.", + "\\end{description}", + "In each case the two given values of $p$ differ in sign,", + " so $\\mathbf{X}_0$ is not a local extreme point of $f$, from", + "Theorem~\\ref{thmtype:5.4.10}\\part{a}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:5.4.10", + "TRENCH_REAL_ANALYSIS-thmtype:5.4.10" + ], + "ref_ids": [ + 167, + 167 + ] + } + ], + "ref_ids": [] + }, + { + "id": 293, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.5", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{F}$ is continuously differentiable on a", + "neighborhood of $\\mathbf{X}_0$ and $J\\mathbf{F}(\\mathbf{X}_0)\\ne 0,$ then", + "there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which the", + "conclusions of Theorem~$\\ref{thmtype:6.3.4}$ hold$.$" + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.3.4" + ], + "proofs": [ + { + "contents": [ + "By continuity, since $J\\mathbf{F}'(\\mathbf{X}_0)\\ne0$,", + " $J\\mathbf{F}'(\\mathbf{X})$", + " is nonzero for all $\\mathbf{X}$ in some open neighborhood $S$ of", + "$\\mathbf{X}_0$. Now apply Theorem~\\ref{thmtype:6.3.4}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:6.3.4" + ], + "ref_ids": [ + 187 + ] + } + ], + "ref_ids": [ + 187 + ] + }, + { + "id": 294, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.4.2", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f:\\R^{n+1}\\to \\R$ is continuously", + "differentiable on an open set containing $(\\mathbf{X}_0,u_0),$ with", + "$f(\\mathbf{X}_0,u_0)=0$", + "and", + "$f_u(\\mathbf{X}_0,u_0)\\ne0$.", + "Then there is a neighborhood $M$ of $(\\mathbf{X}_0,u_0),$ contained in", + "$S,$ and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which", + "is defined a unique continuously differentiable function", + "$u=u(\\mathbf{X}):\\R^n\\to", + "\\R$ such that", + "$$", + "(\\mathbf{X},u(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}", + " f_u(\\mathbf{X},u(\\mathbf{X}))\\ne0,\\quad\\mathbf{X}\\in N,", + "$$", + "$$", + "u(\\mathbf{X}_0)=u_0, \\mbox{\\quad and \\quad}", + "f(\\mathbf{X},u(\\mathbf{X}))=0,\\quad\\mathbf{X}\\in N.", + "$$", + "The partial derivatives of $u$ are given by", + "$$", + "u_{x_i}(\\mathbf{X})=-\\frac{f_{x_i}(\\mathbf{X},u(\\mathbf{X}))}{", + "f_u(\\mathbf{X},u(\\mathbf{X}))},\\quad 1\\le i\\le n.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "We will show that if $f$ is unbounded on $R$, ${\\bf", + "P}=\\{R_1,R_2, \\dots,R_k\\}$ is", + "any partition of $R$, and $M>0$, then there are Riemann sums $\\sigma$", + "and $\\sigma'$ of $f$ over ${\\bf P}$ such that", + "\\begin{equation} \\label{eq:7.1.11}", + "|\\sigma-\\sigma'|\\ge M.", + "\\end{equation}", + "This implies that", + "$f$ cannot satisfy Definition~\\ref{thmtype:7.1.2}. (Why?)", + "Let", + "$$", + "\\sigma=\\sum_{j=1}^kf(\\mathbf{X}_j)V(R_j)", + "$$", + "be a Riemann sum of $f$ over ${\\bf P}$. There must be", + "an integer $i$ in $\\{1,2, \\dots,k\\}$ such that", + "\\begin{equation} \\label{eq:7.1.12}", + "|f(\\mathbf{X})-f(\\mathbf{X}_i)|\\ge\\frac{M }{ V(R_i)}", + "\\end{equation}", + "for some $\\mathbf{X}$ in $R_i$, because if this were not so, we", + "would have", + "$$", + "|f(\\mathbf{X})-f(\\mathbf{X}_j)|<\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", + "\\quad 1\\le j\\le k.", + "$$", + "If this is so, then", + "\\begin{eqnarray*}", + "|f(\\mathbf{X})|\\ar=|f(\\mathbf{X}_j)+f(\\mathbf{X})-f(\\mathbf{X}_j)|\\le|f(\\mathbf{X}_j)|+|f(\\mathbf{X})-f(\\mathbf{X}_j)|\\\\", + "\\ar\\le |f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)},\\quad \\mathbf{X}\\in R_j,\\quad", + "1\\le j\\le k.", + "\\end{eqnarray*}", + "However, this implies that", + "$$", + "|f(\\mathbf{X})|\\le\\max\\set{|f(\\mathbf{X}_j)|+\\frac{M}{ V(R_j)}}{1\\le j\\le k},", + "\\quad \\mathbf{X}\\in R,", + "$$", + "which contradicts the assumption that $f$ is unbounded on $R$.", + " Now suppose that $\\mathbf{X}$ satisfies \\eqref{eq:7.1.12}, and", + "consider the Riemann sum", + "$$", + "\\sigma'=\\sum_{j=1}^nf(\\mathbf{X}_j')V(R_j)", + "$$", + "over the same partition ${\\bf P}$, where", + "$$", + "\\mathbf{X}_j'=\\left\\{\\casespace\\begin{array}{ll}", + "\\mathbf{X}_j,&j \\ne i,\\\\", + "\\mathbf{X},&j=i.\\end{array}\\right.", + "$$", + "Since", + "$$", + "|\\sigma-\\sigma'|=|f(\\mathbf{X})-f(\\mathbf{X}_i)|V(R_i),", + "$$", + "\\eqref{eq:7.1.12} implies \\eqref{eq:7.1.11}." + ], + "refs": [ + "TRENCH_REAL_ANALYSIS-thmtype:7.1.2" + ], + "ref_ids": [ + 359 + ] + } + ], + "ref_ids": [] + }, + { + "id": 295, + "type": "theorem", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.31", + "categories": [], + "title": "", + "contents": [ + "Suppose that", + " $f$ is integrable on sets $S_1$ and $S_2$ such that $S_1\\cap S_2$", + "has zero content$.$ Then $f$ is integrable on $S_1\\cup S_2,$ and", + "$$", + "\\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+", + "\\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "proofs": [ + { + "contents": [ + "Let", + "$$", + "P_1: a=x_00$ a", + "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", + "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from", + "\\eqref{eq:7.2.6}, there is", + "a partition $P_2$ of $[c,d]$ such that", + "$S_F(P_2)-s_F(P_2)<\\epsilon$,", + " so $F$ is integrable on $[c,d]$, from", + "Theorem~\\ref{thmtype:3.2.7}.", + "It remains to verify \\eqref{eq:7.2.1}. From \\eqref{eq:7.2.4} and the", + "definition of $\\int_c^dF(y)\\,dy$,", + "there is for each $\\epsilon>0$ a $\\delta>0$ such that", + "$$", + "\\left|\\int_c^d F(y)\\,dy-\\sigma\\right|<\\epsilon\\mbox{\\quad if\\quad}", + "\\|P_2\\|<\\delta;", + "$$", + "that is,", + "$$", + "\\sigma-\\epsilon<\\int_c^d F(y)\\,dy<\\sigma+\\epsilon\\mbox{\\quad if \\quad}", + "\\|P_2\\|<\\delta.", + "$$", + "This and \\eqref{eq:7.2.5} imply that", + "$$", + "s_f(\\mathbf{P})-\\epsilon<\\int_c^d F(y)\\,dy0$ a", + "partition $\\mathbf{P}$ of $R$ such that $S_f(\\mathbf{P})-s_f(\\mathbf{P})<\\epsilon$,", + "from Theorem~\\ref{thmtype:7.1.12}. Consequently, from \\eqref{eq:7.2.11},", + "there", + "is a partition $\\mathbf{Q}$ of $T$ such that", + "$S_{F_p}(\\mathbf{Q})-s_{F_p}(\\mathbf{Q})<\\epsilon$, so $F_p$ is integrable", + "on $T$, from Theorem~\\ref{thmtype:7.1.12}.", + "It remains to verify that", + "\\begin{equation} \\label{eq:7.2.12}", + "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=", + "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}.", + "\\end{equation}", + "From \\eqref{eq:7.2.9} and the definition of $\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}$, there", + "is for each $\\epsilon>0$ a $\\delta>0$ such that", + "$$", + "\\left|\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "-\\sigma\\right|<\\epsilon\\mbox{\\quad", + "if\\quad}", + "\\|\\mathbf{Q}\\|<\\delta;", + "$$", + "that is,", + "$$", + "\\sigma-\\epsilon<\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "<\\sigma+", + "\\epsilon\\mbox{\\quad if \\quad}\\|\\mathbf{Q}\\|<\\delta.", + "$$", + "This and \\eqref{eq:7.2.10} imply that", + "$$", + "s_f(\\mathbf{P})-\\epsilon<", + "\\int_TF_p(\\mathbf{Y})\\,d\\mathbf{Y}", + "0$ such that if ${\\bf", + "P}$ is any partition of $C$ with $\\|{\\bf P}\\|\\le\\delta$ and $\\sigma$", + "is any Riemann sum of $\\psi_K$ over ${\\bf P}$, then", + "\\begin{equation}\\label{eq:7.3.6}", + "0\\le\\sigma\\le\\epsilon.", + "\\end{equation}", + "\\newpage", + "\\noindent", + "Now suppose that ${\\bf P}=\\{C_1,C_2,\\dots,C_k\\}$ is a partition of $C$", + "into cubes with", + "\\begin{equation}\\label{eq:7.3.7}", + "\\|{\\bf P}\\|<\\min (\\rho,\\delta),", + "\\end{equation}", + "and let $C_1$, $C_2$, \\dots, $C_k$ be numbered so that $C_j\\cap K\\ne", + "\\emptyset$ if $1\\le j\\le r$ and", + "$C_j\\cap K=\\emptyset$ if $r+1\\le j\\le k$. Then \\eqref{eq:7.3.5} holds, and", + "a typical Riemann sum of $\\psi_K$ over ${\\bf P}$ is of the form", + "$$", + "\\sigma=\\sum_{j=1}^r\\psi_K(\\mathbf{X}_j)V(C_j)", + "$$", + "with $\\mathbf{X}_j\\in C_j$, $1\\le j\\le r$. In particular, we", + "can choose", + "$\\mathbf{X}_j$ from $K$, so that $\\psi_K(\\mathbf{X}_j)=1$, and", + "$$", + "\\sigma=\\sum_{j=1}^r V(C_j).", + "$$", + "Now \\eqref{eq:7.3.6} and \\eqref{eq:7.3.7} imply that $C_1$, $C_2$, \\dots,", + "$C_r$ have the required properties." + ], + "refs": [], + "ref_ids": [] + } + ], + "ref_ids": [] + } + ], + "definitions": [ + { + "id": 298, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.1.5", + "categories": [], + "title": "", + "contents": [ + "A set $D$ is {\\it dense in the reals\\/}", + "if every open interval $(a,b)$ contains a member of $D$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 299, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.1", + "categories": [], + "title": "", + "contents": [ + " Let $S$ and $T$ be sets.", + "\\begin{alist}", + "\\item % (a)", + "$S$ {\\it contains\\/} $T$, and we write $S\\supset T$ or $T\\subset", + "S$, if every member of $T$ is also in $S$. In this case, $T$ is", + "a {\\it subset\\/} of $S$.", + "\\item % (b)", + " $S-T$ is the set of elements that are in $S$ but not in $T$.", + "\\item % (c)", + "$S$ {\\it equals\\/} $T$, and we write $S=T$,", + "if", + "$S$ contains", + "$T$ and", + "$T$ contains $S$; thus, $S=T$ if and only if $S$ and $T$ have the same", + "members.", + "\\newpage", + "\\item % (d)", + " $S$ {\\it strictly contains\\/} $T$", + "if $S$ contains $T$ but $T$ does not contain $S$; that", + "is, if every member of $T$ is also in $S$, but at least one member", + "of", + "$S$ is not in $T$ (Figure~\\ref{figure:1.3.1}).", + "\\item % (e)", + "The {\\it complement\\/} of $S$, denoted by $S^c$,", + "is the set of elements in the universal set that are not in $S$.", + "\\item % (f)", + " The {\\it union\\/} of $S$", + "and", + "$T$, denoted by", + "$S\\cup T$, is the set of elements in at least one of $S$ and $T$", + "(Figure~\\ref{figure:1.3.1}\\part{b}).", + "\\item % (g)", + "The {\\it intersection\\/} of $S$ and $T$, denoted by", + "$S\\,\\cap\\, T$, is the", + "set of elements in both $S$ and $T$ (Figure~\\ref{figure:1.3.1}\\part{c}).", + "If $S\\cap T=\\emptyset$ (the empty set), then $S$ and $T$ are", + " {\\it disjoint sets\\/}", + "(Figure~\\ref{figure:1.3.1}\\part{d}).", + "\\item % (h)", + " A set with only one member $x_0$ is a {\\it singleton", + "set\\/}, denoted by", + "$\\{x_0\\}$.", + "\\end{alist}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 300, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.2", + "categories": [], + "title": "", + "contents": [ + "If $x_0$ is a real number and $\\epsilon>0$, then the open interval", + "$(x_0-\\epsilon, x_0+\\epsilon)$ is an {\\it $\\epsilon$-neighborhood\\/}", + "of", + "$x_0$.", + "If a set $S$ contains an $\\epsilon$-neighborhood of $x_0$, then $S$ is a", + "{\\it neighborhood\\/} of $x_0$, and $x_0$ is an {\\it interior point\\/} of", + "$S$ (Figure~\\ref{figure:1.3.2}). The set of interior points of $S$ is the", + "{\\it interior\\/} of $S$, denoted by $S^0$. If every point of $S$ is an", + "interior point (that is, $S^0=S$), then $S$ is {\\it open\\/}.", + " A set $S$ is \\emph{closed} if $S^c$ is open." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 301, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:1.3.4", + "categories": [], + "title": "", + "contents": [ + "R}$. Then", + "\\begin{alist}", + "\\item % (a)", + " $x_0$ is a {\\it limit point\\/}", + "of $S$ if every deleted neighborhood of $x_0$ contains a point of~$S$.", + "\\item % (b)", + "$x_0$ is a {\\it boundary point\\/} of $S$ if every neighborhood", + "of $x_0$ contains at least one point in $S$ and one not in $S$. The set of", + "boundary points of $S$ is the {\\it boundary\\/} of $S$, denoted by $\\partial", + "S$. The {\\it closure\\/} of $S$, denoted by $\\overline{S}$, is", + "$\\overline{S}=S\\cup \\partial S$.", + "\\item % (c)", + "$x_0$ is an \\emph{isolated point} of $S$ if $x_0\\in S$", + " and there is a neighborhood of $x_0$ that contains no other point of", + "$S$.", + "\\item % (d)", + "$x_0$ is \\emph{exterior} to $S$ if $x_0$ is in the interior of $S^c$. The", + "collection of such points is the {\\it exterior\\/} of $S$.", + "\\end{alist}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 302, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.1", + "categories": [], + "title": "", + "contents": [ + "If $D_f\\cap D_g\\ne", + "\\emptyset,$ then $f+g,$ $f-g,$ and $fg$ are defined on", + "$D_f\\cap D_g$ by", + "\\begin{eqnarray*}", + "(f+g)(x)\\ar= f(x)+g(x),\\\\", + "(f-g)(x)\\ar= f(x)-g(x),\\\\", + "\\noalign{\\hbox{and}}", + "(fg)(x)\\ar= f(x)g(x).", + "\\end{eqnarray*}", + "The quotient $f/g$ is defined by", + "$$", + "\\left(\\frac{f}{ g}\\right) (x)=\\frac{f(x)}{ g(x)}", + "$$", + "for $x$ in $D_f\\cap D_g$ such that $g(x)\\ne0.$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 303, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.2", + "categories": [], + "title": "", + "contents": [ + " We say that $f(x)$ {\\it approaches the limit $L$ as $x$ approaches\\/}", + "$x_0$, and write", + "$$", + "\\lim_{x\\to x_0} f(x)=L,", + "$$", + "if $f$ is defined on some deleted neighborhood of $x_0$ and, for", + "every $\\epsilon>0$, there is a $\\delta>0$ such that", + "\\begin{equation}\\label{eq:2.1.4}", + "|f(x)-L|<\\epsilon", + "\\end{equation}", + "if", + "\\begin{equation}\\label{eq:2.1.5}", + "0<|x-x_0|<\\delta.", + "\\end{equation}", + "Figure~\\ref{figure:2.1.1} depicts the graph", + "of a function for which", + "$\\lim_{x", + "\\to x_0}f(x)$ exists." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 304, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.5", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + "We say that $f(x)$ {\\it approaches the left-hand limit $L$ as", + "$x$ approaches $x_0$ from the left\\/}, and write", + "$$", + "\\lim_{x\\to x_0-} f(x)=L,", + "$$", + "if $f$ is defined on some open interval $(a,x_0)$ and, for each", + "$\\epsilon>0$, there is a $\\delta>0$ such that", + "$$", + "|f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_0-\\delta0$, there is a $\\delta>0$ such that", + "$$", + "|f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_00$, there is a number $\\beta$ such that", + "$$", + "|f(x)-L|<\\epsilon\\quad\\mbox{\\quad if \\quad} x>\\beta.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 306, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.1.8", + "categories": [], + "title": "", + "contents": [ + "We say that $f(x)$ {\\it approaches $\\infty$ as $x$ approaches $x_0$", + "from the left\\/}, and write", + "$$", + "\\lim_{x\\to x_0-} f(x)=\\infty\\mbox{\\quad or \\quad} f(x_0-)=\\infty,", + "$$", + "if $f$ is defined on an interval $(a,x_0)$ and, for each real number", + "$M$, there is a $\\delta>0$ such that", + "$$", + "f(x)>M\\mbox{\\quad if \\quad} x_0-\\delta0$, there is a $\\delta>0$ such", + "that", + "$$", + "|f(x)-f(x')|<\\epsilon\\mbox{\\ whenever }\\ |x-x'|<\\delta", + "\\mbox{\\ and }\\ x,x'\\in S.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 313, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.1", + "categories": [], + "title": "", + "contents": [ + "A function $f$ is {\\it differentiable\\/}", + "at an interior point $x_0$ of its domain if the difference quotient", + "$$", + "\\frac{f(x)-f(x_0)}{ x-x_0},\\quad x\\ne x_0,", + "$$", + "approaches a limit as $x$ approaches $x_0$, in which case the limit is", + "called the {\\it derivative of $f$ at $x_0$\\/}, and", + "is denoted by", + "$f'(x_0)$; thus,", + "\\begin{equation}\\label{eq:2.3.1}", + "f'(x_0)=\\lim_{x\\to x_0}\\frac{f(x)-f(x_0)}{ x-x_0}.", + "\\end{equation}", + "It is sometimes convenient to let $x=x_0+h$ and write \\eqref{eq:2.3.1}", + "as", + "$$", + "f'(x_0)=\\lim_{h\\to 0}\\frac{f(x_0+h)-f(x_0)}{ h}.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 314, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:2.3.6", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + "We say that $f$ is {\\it differentiable on the closed interval\\/}", + "$[a,b]$ if $f$ is differentiable on the open interval $(a,b)$ and", + "$f_+'(a)$ and $f_-'(b)$ both exist.", + "\\item % (b)", + "We say that $f$ is {\\it continuously differentiable on\\/}", + "$[a,b]$ if $f$ is differentiable on $[a,b]$, $f'$ is continuous", + "on $(a,b)$,", + "$f_+'(a)=f'(a+)$, and $f_-'(b)=f'(b-)$.", + "\\end{alist}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 315, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.1", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be defined on $[a,b]$. We say that $f$ is", + "{\\it Riemann integrable on\\/}", + "$[a,b]$ if there", + "is a number $L$ with the following property: For every $\\epsilon>0$,", + "there is a $\\delta>0$ such that", + "$$", + "\\left|\\sigma-L \\right|<\\epsilon", + "$$", + "if $\\sigma$ is any Riemann sum of $f$ over", + "a partition $P$ of $[a,b]$", + "such that $\\|P\\|<\\delta$.", + "In this case, we say that $L$ is {\\it the Riemann integral of", + "$f$ over\\/} $[a,b]$,", + "and write", + "$$", + "\\int_a^b f(x)\\,dx=L.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 316, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.3", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on $[a,b]$ and", + "$P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, let", + "\\begin{eqnarray*}", + "M_j\\ar=\\sup_{x_{j-1}\\le x\\le x_j}f(x)\\\\", + "\\arraytext{and}\\\\", + "m_j\\ar=\\inf_{x_{j-1}\\le x\\le x_j}f(x).", + "\\end{eqnarray*}", + "The {\\it upper sum of $f$ over $P$\\/}", + " is", + "$$", + "S(P)=\\sum_{j=1}^n M_j(x_j-x_{j-1}),", + "$$", + "and the {\\it upper integral of $f$ over\\/},", + "$[a,b]$, denoted by", + "$$", + "\\overline{\\int_a^b} f(x)\\,dx,", + "$$", + "is the infimum of all upper sums. The {\\it lower", + "sum of $f$ over $P$\\/}", + "is", + "$$", + "s(P)=\\sum_{j=1}^n m_j(x_j-x_{j-1}),", + "$$", + "and the {\\it lower integral of $f$ over\\/}", + "$[a,b]$, denoted by", + "$$", + "\\underline{\\int_a^b}f(x)\\,dx,", + "$$", + "is the supremum of all lower sums.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 317, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.1.5", + "categories": [], + "title": "", + "contents": [ + "Let $f$ and $g$ be defined on $[a,b]$. We say that $f$ is", + "{\\it Riemann}--\\href{http://www-history.mcs.st-and.ac.uk/Mathematicians/Stieltjes.html}", + "{\\it Stieltjes}", + "{\\it integrable with respect to $g$ on\\/}", + "$[a,b]$", + "if there", + "is a number $L$ with the following property: For every $\\epsilon>0$,", + "there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:3.1.15}", + "\\left|\\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right]-L \\right|<", + "\\epsilon,", + "\\end{equation}", + "provided only that $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$", + "such that $\\|P\\|<\\delta$ and", + "$$", + "x_{j-1}\\le c_j\\le x_j,\\quad j=1,2, \\dots,n.", + "$$", + "In this case, we say that $L$ is {\\it the Riemann--Stieltjes integral", + "of", + "$f$ with respect to $g$ over\\/}", + "$[a,b]$, and write", + "$$", + "\\int_a^b f(x)\\,dg (x)=L.", + "$$", + "The sum", + "$$", + "\\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right]", + "$$", + "in \\eqref{eq:3.1.15} is {\\it a Riemann--Stieltjes sum of $f$", + "with respect to $g$ over the partition~$P$\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 318, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.1", + "categories": [], + "title": "", + "contents": [ + "If $f$ is locally integrable on", + "$[a,b)$, we define", + "\\begin{equation}\\label{eq:3.4.1}", + "\\int_a^b f(x)\\,dx=", + "\\lim_{c\\to b-}\\int_a^c f(x)\\,dx", + "\\end{equation}", + "if the limit exists (finite). To include the case where $b=\\infty$, we", + "adopt the convention that $\\infty-=\\infty$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 319, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.2", + "categories": [], + "title": "", + "contents": [ + " If $f$ is locally integrable on", + "$(a,b]$, we define", + "$$", + "\\int_a^b f(x)\\,dx=\\lim_{c\\to a+}\\int_c^b f(x)\\,dx", + "$$", + "provided that the limit exists (finite).", + " To include the case where $a=-\\infty$, we adopt the", + "convention that $-\\infty+=-\\infty$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 320, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:3.4.3", + "categories": [], + "title": "", + "contents": [ + "If $f$ is locally integrable on", + "$(a,b),$ we define", + "$$", + "\\int_a^b f(x)\\,dx=\\int_a^\\alpha f(x)\\,dx+\\int_\\alpha^b f(x)\\,dx,", + "$$", + "where $a<\\alpha0$ there is a finite or infinite sequence of", + "open intervals $I_1$, $I_2$, \\dots\\ such that", + "\\begin{equation} \\label{eq:3.5.8}", + "S\\subset\\bigcup_j I_j", + "\\end{equation}", + "and", + "\\begin{equation} \\label{eq:3.5.9}", + "\\sum_{j=1}^n L(I_j)<\\epsilon,\\quad n\\ge1.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 324, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.1", + "categories": [], + "title": "", + "contents": [ + "A sequence $\\{s_n\\}$ {\\it converges to a limit $s$\\/} if for", + "every $\\epsilon>0$ there is an integer $N$ such that", + "\\begin{equation}\\label{eq:4.1.2}", + "|s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", + "\\end{equation}", + "In this case we say that $\\{s_n\\}$ is {\\it convergent\\/} and write", + "$$", + "\\lim_{n\\to\\infty}s_n=s.", + "$$", + "A sequence that does not converge {\\it diverges\\/}, or is", + "{\\it divergent\\/}", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 325, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.3", + "categories": [], + "title": "", + "contents": [ + "A sequence $\\{s_n\\}$ is {\\it bounded above\\/}", + " if there is a real number $b$ such that", + "$$", + "s_n\\le b\\mbox{\\quad for all $n$},", + "$$", + "{\\it bounded below\\/} if there is a", + "real number", + "$a$ such that", + "$$", + "s_n\\ge a\\mbox{\\quad for all $n$},", + "$$", + "or {\\it bounded\\/} if", + "there is a real number", + "$r$ such that", + "$$", + "|s_n|\\le r\\mbox{\\quad for all $n$}.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 326, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.1.5", + "categories": [], + "title": "", + "contents": [ + " A sequence $\\{s_n\\}$ is {\\it nondecreasing\\/} if", + "$s_n\\ge", + "s_{n-1}$ for all $n$, or {\\it nonincreasing\\/} if", + "$s_n\\le s_{n-1}$", + "for all $n.$ A {\\it monotonic sequence\\/}", + "is a sequence that is either", + "nonincreasing or nondecreasing. If $s_n>s_{n-1}$ for all $n$, then", + "$\\{s_n\\}$ is {\\it increasing\\/},", + "while if", + "$s_n 0$ there is an integer $N$ such that", + "\\begin{equation} \\label{eq:4.4.1}", + "\\|F_n-F\\|_S<\\epsilon\\mbox{\\quad if\\quad} n\\ge N.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 334, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.4.12", + "categories": [], + "title": "", + "contents": [ + "If $\\{f_j\\}^\\infty_k$ is a sequence of real-valued functions defined", + "on a set $D$ of reals, then $\\sum_{j=k}^\\infty f_j$ is an", + "{\\it infinite series\\/} (or simply a {\\it", + "series\\/}) of functions on", + "$D$. The {\\it partial sums of\\/},", + "$\\sum_{j=k}^\\infty f_j$ are defined by", + "$$", + "F_n=\\sum^n_{j=k} f_j,\\quad n\\ge k.", + "$$", + "If $\\{F_n\\}^\\infty_k$ converges pointwise to a function $F$ on a", + "subset $S$ of $D$, we say that $\\sum_{j=k}^\\infty f_j$ {\\it converges", + "pointwise to the sum $F$ on\\/} $S$, and write", + "$$", + "F=\\sum_{j=k}^\\infty f_j,\\quad x\\in S.", + "$$", + "\\newpage", + "\\noindent", + "If $\\{F_n\\}$ converges uniformly to $F$ on $S$, we say that", + "$\\sum_{j=k}^\\infty f_j$ {\\it converges uniformly to $F$ on~$S$\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 335, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:4.5.1", + "categories": [], + "title": "", + "contents": [ + "An infinite series of the form", + "\\begin{equation}\\label{eq:4.5.1}", + "\\sum^\\infty_{n=0} a_n(x-x_0)^n,", + "\\end{equation}", + "where $x_0$ and $a_0$, $a_1$, \\dots, are constants, is called a {\\it", + "power series in $x-x_0$\\/}.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 336, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.1", + "categories": [], + "title": "", + "contents": [ + "The {\\it vector sum\\/} of", + "$$", + "\\mathbf{X}=(x_1,x_2, \\dots,x_n)\\mbox{\\quad and\\quad}\\mathbf{Y}=", + "(y_1,y_2, \\dots,y_n)", + "$$", + "is", + "\\begin{equation}\\label{eq:5.1.1}", + "\\mathbf{X}+\\mathbf{Y}=(x_1+y_1,x_2+y_2, \\dots,x_n+y_n).", + "\\end{equation}", + "If $a$ is a real number, the {\\it scalar multiple of $\\mathbf{X\\/}$ by\\/}", + "$a$ is", + "\\begin{equation}\\label{eq:5.1.2}", + "a\\mathbf{X}=(ax_1,ax_2, \\dots,ax_n).", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 337, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.3", + "categories": [], + "title": "", + "contents": [ + "The {\\it length\\/} of the vector", + "$\\mathbf{X}=(x_1,x_2, \\dots, x_n)$ is", + "$$", + "|\\mathbf{X}|=(x^2_1+x^2_2+\\cdots+x^2_n)^{1/2}.", + "$$", + "The {\\it distance between points $\\mathbf{X\\/}$ and\\/} $\\mathbf{Y}$ is", + "$|\\mathbf{X}-\\mathbf{Y}|$; in particular, $|\\mathbf{X}|$ is the distance between", + "$\\mathbf{X}$ and the origin. If $|\\mathbf{X}|=1$, then $\\mathbf{X}$ is", + "a {\\it unit vector\\/}.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 338, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.4", + "categories": [], + "title": "", + "contents": [ + "The {\\it inner product\\/} $\\mathbf{X}\\cdot", + "\\mathbf{Y}$ of $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$ and $\\mathbf{Y}=", + "(y_1,y_2, \\dots,y_n)$ is", + "$$", + "\\mathbf{X}\\cdot\\mathbf{Y}=x_1y_1+x_2y_2+\\cdots+x_ny_n.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 339, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.10", + "categories": [], + "title": "", + "contents": [ + "$\\mathbf{U}$ are in $\\R^n$ and $\\mathbf{U}\\ne\\mathbf{0}$. Then {\\it the", + "line through $\\mathbf{X}_0$ in the direction of\\/}", + "$\\mathbf{U}$ is the set of all points in $\\R^n$ of the form", + "$$", + "\\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U},\\quad -\\infty0$, the {\\it $\\epsilon$-neighborhood of a point\\/}", + "$\\mathbf{X}_{0}$ in", + "$\\R^n$ is the set", + "$$", + "N_\\epsilon(\\mathbf{X}_0)|=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\epsilon}.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 341, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.13", + "categories": [], + "title": "", + "contents": [ + "A sequence of points $\\{\\mathbf{X}_r\\}$ in $\\R^n$", + "{\\it converges to the limit\\/} $\\overline{\\mathbf{X}}$ if", + "$$", + "\\lim_{r\\to\\infty} |\\mathbf{X}_r-\\overline{\\mathbf{X}}|=0.", + "$$", + "In this case we write", + "$$", + "\\lim_{r\\to\\infty}\\mathbf{X}_r=\\overline{\\mathbf{X}}.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 342, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.16", + "categories": [], + "title": "", + "contents": [ + "If $S$ is a nonempty subset of $\\R^n$, then", + "$$", + "d(S)=\\sup\\set{|\\mathbf{X}-\\mathbf{Y}|}{\\mathbf{X},\\mathbf{Y}\\in S}", + "$$", + "is the {\\it diameter\\/} of $S$.", + "If $d(S)<\\infty,$ $S$ is {\\it bounded\\/}$;$ if", + "$d(S)=\\infty$, $S$ is {\\it unbounded\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 343, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.19", + "categories": [], + "title": "", + "contents": [ + "A subset $S$ of $\\R^n$ is", + " {\\it connected\\/} if it is impossible to represent", + "$S$ as the union of two", + "disjoint nonempty sets such that neither contains a limit point of the", + "other; that is, if $S$ cannot be expressed as $S=A\\cup B$, where", + "\\begin{equation}\\label{eq:5.1.16}", + "A\\ne\\emptyset,\\quad B\\ne\\emptyset,\\quad\\overline{A}\\cap B=", + "\\emptyset,\\mbox{\\quad and\\quad} A\\cap\\overline{B}=\\emptyset.", + "\\end{equation}", + "If $S$ can be expressed in this way, then $S$ is", + "{\\it disconnected\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 344, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.1.21", + "categories": [], + "title": "", + "contents": [ + "A {\\it region\\/} $S$ in $\\R^n$ is the union of an open connected", + "set", + "with some, all, or none of its boundary; thus, $S^0$ is connected, and", + "every point of $S$ is a limit point of $S^0$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 345, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.1", + "categories": [], + "title": "", + "contents": [ + "We say that $f(\\mathbf{X})$", + "{\\it approaches the limit $L$ as $\\mathbf{X\\/}$ approaches\\/} $\\mathbf{X}_0$", + "and write", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L", + "$$", + "if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every $\\epsilon>0$,", + "there is a $\\delta>0$ such that", + "$$", + "|f(\\mathbf{X})-L|<\\epsilon", + "$$", + "for all $\\mathbf{X}$ in $D_f$ such that", + "$$", + "0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 346, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.4", + "categories": [], + "title": "", + "contents": [ + "We say that $f(\\mathbf{X})$ {\\it approaches $\\infty$ as $\\mathbf{X\\/}$", + "approaches", + "$\\mathbf{X}_0$\\/} and write", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=\\infty", + "$$", + "if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every real number", + "$M$, there is a $\\delta>0$ such that", + "$$", + "f(\\mathbf{X})>M\\mbox{\\quad whenever\\quad} 0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta", + "\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", + "$$", + "We say that", + "\\begin{eqnarray*}", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})\\ar=-\\infty\\\\", + "\\arraytext{if}\\\\", + "\\lim_{{\\mathbf{X}}\\to\\mathbf{X}_0} (-f)(\\mathbf{X})\\ar=\\infty.", + "\\end{eqnarray*}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 347, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.5", + "categories": [], + "title": "", + "contents": [ + "If $D_f$ is unbounded$,$ we say that", + "$$", + "\\lim_{|\\mathbf{X}|\\to\\infty} f(\\mathbf{X})=L\\mbox{\\quad (finite)\\quad}", + "$$", + "if for every $\\epsilon>0$, there is a number $R$ such that", + "$$", + "|f(\\mathbf{X})-L|<\\epsilon\\mbox{\\quad whenever\\quad}\\ |\\mathbf{X}|\\ge R", + "\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 348, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.2.6", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f$, then we say", + "that $f$ is", + "{\\it continuous at $\\mathbf{X\\/}_0$\\/} if", + "$$", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=f(\\mathbf{X}_0).", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 349, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.1", + "categories": [], + "title": "", + "contents": [ + "Let $\\boldsymbol{\\Phi}$ be a unit vector and $\\mathbf{X}$ a point in", + "$\\R^n$.", + " {\\it The directional derivative of $f$ at $\\mathbf{X}$ in the", + "direction of\\/} $\\boldsymbol{\\Phi}$ is defined by", + "$$", + "\\frac{\\partial f(\\mathbf{X})}{\\partial\\boldsymbol{\\Phi}}=\\lim_{t\\to", + "0}\\frac", + "{f(\\mathbf{X}+ t\\boldsymbol{\\Phi})-f(\\mathbf{X})}{ t}", + "$$", + "if the limit exists. That is, $\\partial f(\\mathbf{X})/\\partial\\boldsymbol{\\Phi}$", + "is the ordinary derivative of the function", + "$$", + "h(t)=f(\\mathbf{X}+t\\boldsymbol{\\Phi})", + "$$", + "at $t=0$, if $h'(0)$ exists." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 350, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.3.5", + "categories": [], + "title": "", + "contents": [ + "A function $f$ is {\\it differentiable\\/} at", + "$$", + " \\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0}))", + "$$", + "if $\\mathbf{X}_0\\in D_f^0$ and", + "there are constants $m_1$, $m_2$, \\dots$,$ $m_n$ such that", + "\\begin{equation}\\label{eq:5.3.16}", + "\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)-", + "\\dst{\\sum^n_{i=1}}\\, m_i (x_i-x_{i0})}{ |\\mathbf{X}-\\mathbf{X}_0|}=0.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 351, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.1", + "categories": [], + "title": "", + "contents": [ + "A vector-valued function", + " $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is {\\it", + "differentiable\\/} at", + "$$", + "\\mathbf{U}_0=(u_{10},u_{20}, \\dots,u_{m0})", + "$$", + " if its component functions", + "$g_1$, $g_2$, \\dots, $g_n$ are differentiable at $\\mathbf{U}_0$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 352, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:5.4.7", + "categories": [], + "title": "", + "contents": [ + "Suppose that $r\\ge1$ and all partial derivatives of $f$ of order $\\le r-1$", + "are differentiable in a neighborhood of $\\mathbf{X}_0$. Then the $r$th", + "{\\it differential of $f$ at\\/} $\\mathbf{X}_0$, denoted by $d^{(r)}_{\\mathbf{X}_0}f$, is defined by", + "\\begin{equation} \\label{eq:5.4.23}", + "d^{(r)}_{\\mathbf{X}_0}f=\\sum_{i_1,i_2, \\dots,i_r=1}^n", + "\\frac{\\partial^rf(\\mathbf{X}_0)", + "}{\\partial x_{i_r}\\partial x_{i_{r-1}}\\cdots\\partial x_{i_1}}", + "dx_{i_1}dx_{i_2}\\cdots dx_{i_r},", + "\\end{equation}", + "where $dx_1$, $dx_2$, \\dots, $dx_n$ are the differentials", + "introduced in Section~5.3; that is, $dx_i$ is the function", + "whose value at a point in $\\R^n$ is the $i$th coordinate", + "of the point.", + "For convenience, we define", + "$$", + "(d^{(0)}_{\\mathbf{X}_0}f)=f(\\mathbf{X}_0).", + "$$", + "Notice that $d^{(1)}_{\\mathbf{X}_0}f=d_{\\mathbf{X}_0}f$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 353, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.1", + "categories": [], + "title": "", + "contents": [ + "A transformation $\\mathbf{L}: \\R^n \\to \\R^m$", + "defined on all of", + "$\\R^n$ is {\\it linear\\/} if", + "$$", + "\\mathbf{L}(\\mathbf{X}+\\mathbf{Y})=\\mathbf{L}(\\mathbf{X})+\\mathbf{L}(\\mathbf{Y})", + "$$", + "for all $\\mathbf{X}$ and $\\mathbf{Y}$ in $\\R^n$ and", + "$$", + "\\mathbf{L}(a\\mathbf{X})=a\\mathbf{L}(\\mathbf{X})", + "$$", + "for all $\\mathbf{X}$ in $\\R^n$ and real numbers $a$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 354, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.3", + "categories": [], + "title": "", + "contents": [ + "\\begin{alist}", + "\\item % (a)", + " If $c$ is a real number and", + "$\\mathbf{A}=[a_{ij}]$ is an $m\\times n$ matrix, then $c\\mathbf{A}$ is the", + "$m\\times n$ matrix defined by", + "$$", + "c\\mathbf{A}=[ca_{ij}];", + "$$", + "that is, $c\\mathbf{A}$ is obtained by multiplying every entry of", + "$\\mathbf{A}$ by $c$.", + "\\item % (b)", + "If $\\mathbf{A}=[a_{ij}]$ and $\\mathbf{B}=[b_{ij}]$ are $m\\times n$", + "matrices, then the {\\it sum\\/}", + " $\\mathbf{A}+ \\mathbf{B}$", + " is the", + "$m\\times n$ matrix", + "$$", + "\\mathbf{A}+\\mathbf{B}=[a_{ij}+b_{ij}];", + "$$", + "that is, the sum of two $m\\times n$ matrices is obtained by adding", + "corresponding entries. The sum of two matrices is not defined unless", + "they have the same number of rows and the same number of columns.", + "\\item % (c)", + "If $\\mathbf{A}=[a_{ij}]$ is an $m\\times p$ matrix and $\\mathbf{B}= [b_{ij}]$", + "is a $p\\times n$ matrix, then the {\\it product\\/}", + "$\\mathbf{C}=\\mathbf{AB}$ is the $m\\times n$ matrix with", + "$$", + "c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\\cdots+a_{ip}b_{pj}=\\sum^p_{k=1}", + "a_{ik}b_{kj},\\quad 1\\le i\\le m,\\ 1\\le j\\le n.", + "$$", + "Thus, the $(i,j)$th entry of $\\mathbf{AB}$ is obtained by", + "multiplying each entry in the $i$th row of $\\mathbf{A}$ by the", + "corresponding entry in the $j$th column of $\\mathbf{B}$ and adding the", + "products. This definition requires that $\\mathbf{A}$ have the same number", + "of columns as $\\mathbf{B}$ has rows. Otherwise, $\\mathbf{AB}$ is", + "undefined.", + "\\end{alist}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 355, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.8", + "categories": [], + "title": "", + "contents": [ + "The {\\it norm\\/}$,$ $\\|\\mathbf{A}\\|,$ of an $m\\times n$ matrix", + "$\\mathbf{A}=[a_{ij}]$ is the smallest number such that", + "$$", + "|\\mathbf{AX}|\\le\\|\\mathbf{A}\\|\\,|\\mathbf{X}|", + "$$", + "for all $\\mathbf{X}$ in $\\R^n.$", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 356, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.1.10", + "categories": [], + "title": "", + "contents": [ + "Let $\\mathbf{A}=[a_{ij}]$ be an $n\\times n$ matrix$,$ with $n\\ge2.$", + "The {\\it cofactor\\/} of an entry $a_{ij}$ is", + "$$", + "c_{ij}=(-1)^{i+j}\\det(\\mathbf{A}_{ij}),", + "$$", + "where $\\mathbf{A}_{ij}$ is the $(n-1)\\times(n-1)$ matrix obtained by", + "deleting the $i$th row and $j$th column of $\\mathbf{A}.$", + "The {\\it adjoint\\/} of", + "$\\mathbf{A},$ denoted by", + "$\\adj(\\mathbf{A}),$ is the", + "$n\\times n$ matrix whose $(i,j)$th entry is $c_{ji}.$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 357, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:6.3.2", + "categories": [], + "title": "", + "contents": [ + "A transformation $\\mathbf{F}: \\R^n\\to \\R^n$ is", + "{\\it regular\\/} on an open set $S$ if $\\mathbf{F}$ is one-to-one and", + "continuously", + "differentiable on $S$, and $J\\mathbf{F}(\\mathbf{X})\\ne0$ if $\\mathbf{X}\\in S$.", + "We will also say that $\\mathbf{F}$", + " is regular on an arbitrary set $S$ if", + "$\\mathbf{F}$ is regular on an open set containing $S$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 358, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.1", + "categories": [], + "title": "", + "contents": [ + "A {\\it coordinate rectangle\\/} $R$ in $\\R^n$ is the Cartesian", + "product of $n$ closed intervals; that is,", + "$$", + "R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n].", + "$$", + "The {\\it content\\/} of $R$ is", + "$$", + "V(R)=(b_1-a_1)(b_2-a_2)\\cdots (b_n-a_n).", + "$$", + "The numbers $b_1-a_1$, $b_2-a_2$, \\dots, $b_n-a_n$ are the {\\it edge", + "lengths\\/} of $R$. If", + "they are equal, then", + "$R$ is a", + "{\\it coordinate cube\\/}.", + " If $a_r=b_r$ for some $r$, then $V(R)=0$ and we", + "say that $R$ is {\\it degenerate\\/};", + "otherwise,", + "$R$ is", + "{\\it nondegenerate\\/}.", + " \\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 359, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.2", + "categories": [], + "title": "", + "contents": [ + "Let $f$ be a real-valued function defined", + "on a rectangle $R$ in $\\R^n$. We say that", + " $f$ is {\\it Riemann integrable on\\/} $R$", + " if there is a number $L$ with the following property: For", + "every $\\epsilon>0$, there is a $\\delta>0$ such that", + "$$", + "\\left|\\sigma-L\\right|<\\epsilon", + "$$", + "if $\\sigma$ is any Riemann sum of $f$ over", + "a partition ${\\bf P}$ of $R$", + "such that $\\|{\\bf P}\\|<\\delta$.", + "In this case, we say that", + " $L$ is the {\\it Riemann integral of $f$ over\\/} $R$, and write", + "$$", + "\\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 360, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.4", + "categories": [], + "title": "", + "contents": [ + "If $f$ is bounded on a rectangle $R$ in $\\R^n$ and", + "${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a partition of $R$, let", + "$$", + "M_j=\\sup_{\\mathbf{X}\\in R_j}f(\\mathbf{X}),\\quad m_j=", + "\\inf_{\\mathbf{X}\\in R_j}f(\\mathbf{X}).", + "$$", + "The {\\it upper sum\\/} of $f$ over ${\\bf P}$ is", + "$$", + "S({\\bf P})=\\sum_{j=1}^k M_jV(R_j),", + "$$", + "and the {\\it upper integral", + " of $f$ over\\/} $R$, denoted by", + "$$", + "\\overline{\\int_R}\\,f(\\mathbf{X})\\,d\\mathbf{X},", + "$$", + " is the infimum of all upper", + "sums. The {\\it lower sum of $f$ over\\/} ${\\bf P}$ is", + "$$", + "s({\\bf P})=\\sum_{j=1}^k m_jV(R_j),", + "$$", + "and the {\\it lower integral", + " of $f$ over \\/}$R$, denoted by", + "$$", + "\\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X},", + "$$", + " is the supremum of all lower sums.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 361, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.14", + "categories": [], + "title": "", + "contents": [ + "A subset $E$ of $\\R^n$ has zero content if for each", + "$\\epsilon>0$", + "there is a finite set of rectangles $T_1$, $T_2$, \\dots, $T_m$ such", + "that", + "\\begin{equation}\\label{eq:7.1.24}", + "E\\subset\\bigcup_{j=1}^m T_j", + "\\end{equation}", + "and", + "\\begin{equation}\\label{eq:7.1.25}", + "\\sum_{j=1}^m V(T_j)<\\epsilon.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 362, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.17", + "categories": [], + "title": "", + "contents": [ + "Suppose that $f$ is bounded on a bounded subset of $S$ of", + "$\\R^n$, and let", + "\\begin{equation}\\label{eq:7.1.36}", + "f_S(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f(\\mathbf{X}),&\\mathbf{X}\\in", + "S,\\\\[2\\jot]", + " 0,&\\mathbf{X}\\not\\in S.\\end{array}\\right.", + "\\end{equation}", + "Let $R$ be a rectangle containing $S$.", + "Then {\\it the integral of $f$ over $S$\\/} is defined to be", + "$$", + "\\int_S f(\\mathbf{X})\\,d\\mathbf{X}=\\int_R f_S(\\mathbf{X})\\,d\\mathbf{X}", + "$$", + "if $\\int_R f_S(\\mathbf{X})\\,", + "d\\mathbf{X}$ exists.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 363, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.18", + "categories": [], + "title": "", + "contents": [ + "If $S$ is a bounded subset of $\\R^n$ and", + "the integral $\\int_S\\,d\\mathbf{X}$ (with integrand $f\\equiv1$)", + "exists, we call $\\int_S\\,d\\mathbf{X}$ the {\\it content\\/} (also, {\\it area\\/} if", + "$n=2$ or", + "{\\it volume\\/} if $n=3$) of $S$, and denote it by $V(S)$;", + "thus,", + "$$", + "V(S)=\\int_S\\,d\\mathbf{X}.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 364, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.1.20", + "categories": [], + "title": "", + "contents": [ + "A {\\it differentiable surface\\/} $S$ in $\\R^n\\ (n>1)$ is the", + "image of a", + "compact subset $D$ of $\\R^m$, where $m< n$, under a continuously", + "differentiable transformation $\\mathbf{G}: \\R^m\\to \\R^n$. If", + "$m=1$, $S$ is also called a {\\it differentiable curve\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 365, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:7.3.9", + "categories": [], + "title": "", + "contents": [ + "If $\\mathbf{A}=[a_{ij}]$ is an $n \\times n$ matrix$,$ then", + "$$", + "\\max\\set{\\sum_{j=1}^n |a_{ij}|}{1\\le i\\le n}", + "$$", + "is the {\\it infinity norm of\\/} $A,$ denoted by $\\|A\\|_\\infty$." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 366, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.1", + "categories": [], + "title": "", + "contents": [ + "A {\\it metric space\\/} is a nonempty set $A$ together with", + "a real-valued function $\\rho$ defined on $A\\times A$ such that", + " if $u$, $v$, and $w$", + "are arbitrary members of $A$, then", + "\\begin{alist}", + "\\item % (a)", + "$\\rho(u,v)\\ge 0$, with equality if and only if $u=v$;", + "\\item % (b)", + "$\\rho(u,v)=\\rho(v,u)$;", + "\\item % (c)", + "$\\rho(u,v)\\le\\rho(u,w)+\\rho(w,v)$.", + "\\end{alist}", + "We say that $\\rho$ is a {\\it metric\\/} on $A$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 367, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.2", + "categories": [], + "title": "", + "contents": [ + "A {\\it vector space\\/} $A$", + "is a nonempty set of elements called", + "{\\it vectors\\/} on which two operations, vector", + "addition and scalar multiplication", + "(multiplication by real numbers) are defined, such", + "that the following assertions are true for all $\\mathbf{U}$, $\\mathbf{V}$,", + "and $\\mathbf{W}$ in $A$ and all real numbers $r$ and $s$:\\\\", + "\\phantom{1}1. $\\mathbf{U}+\\mathbf{V}\\in A$;\\\\", + "\\phantom{1}2. $\\mathbf{U}+\\mathbf{V}=\\mathbf{V}+\\mathbf{U}$;\\\\", + "\\phantom{1}3. $\\mathbf{U}+(\\mathbf{V}+\\mathbf{W})=(\\mathbf{U}+\\mathbf{V})+\\mathbf{W}$;\\\\", + "\\phantom{1}4. There is a vector $\\mathbf{0}$ in $A$", + "such that $\\mathbf{U}+\\mathbf{0}=\\mathbf{U}$;\\\\", + "\\phantom{1}5. There is a vector $-\\mathbf{U}$ in $A$", + "such that $\\mathbf{U}+(-\\mathbf{U})=\\mathbf{0}$;\\\\", + "\\phantom{1}6. $r\\mathbf{U}\\in A$;\\\\", + "\\phantom{1}7. $r(\\mathbf{U}+\\mathbf{V})=r\\mathbf{U}+r\\mathbf{V}$;\\\\", + "\\phantom{1}8. $(r+s)\\mathbf{U}=r\\mathbf{U}+s\\mathbf{U}$;\\\\", + "\\phantom{1}9. $r(s\\mathbf{U})=(rs)\\mathbf{U}$; \\\\", + "10. $1\\mathbf{U}=\\mathbf{U}$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 368, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.3", + "categories": [], + "title": "", + "contents": [ + "A {\\it normed vector space\\/}", + " is a vector space", + "$A$ together with a real-valued function $N$ defined on", + "$A$, such that", + " if $u$ and $v$", + "are arbitrary vectors in $A$ and $a$ is a real number, then", + "\\begin{alist}", + "\\item % (a)", + "$N(u)\\ge 0$ with equality if and only if $u=0$;", + "\\item % (b)", + "$N(au)=|a|N(u)$;", + "\\item % (c)", + "$N(u+v)\\le N(u)+N(v)$.", + "\\end{alist}", + "We say that $N$ is a {\\it norm\\/} on", + "$A$, and", + "$(A,N)$ is a {\\it normed vector space\\/}." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 369, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.6", + "categories": [], + "title": "", + "contents": [ + "If $p\\ge 1$ and $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$, let", + "\\begin{equation} \\label{eq:8.1.3}", + "\\|\\mathbf{X}\\|_p", + "=\\left(\\sum_{i=1}^n|x_i|^p\\right)^{1/p}.", + "\\end{equation}", + "The metric induced on $\\R^n$ by this norm is", + "$$", + "\\rho_p(\\mathbf{X},\\mathbf{Y})", + "=\\left(\\sum_{i=1}^n|x_i-y_i|^p\\right)^{1/p}.", + "\\eqno{\\bbox}", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 370, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.10", + "categories": [], + "title": "", + "contents": [ + "If $u_0\\in A$ and $\\epsilon>0$, the set", + "$$", + "N_\\epsilon(u_0)=\\set{u\\in A}{\\rho(u_0,u)<\\epsilon}", + "$$", + "is called an {\\it $\\epsilon$-neighborhood\\/} of $u_0$.", + "(Sometimes we call $S_\\epsilon$ the {\\it open ball of radius", + "$\\epsilon$ centered at $u_0$\\/}.)", + "If a subset $S$ of $A$ contains an $\\epsilon$-neighborhood of $u_0$,", + "then", + "$S$ is a {\\it neighborhood\\/} of", + "$u_0$, and", + "$u_0$ is an", + "{\\it interior point\\/} of", + "$S$. The set of interior points of", + "$S$ is the {\\it interior\\/} of $S$,", + "denoted by", + "$S^0$. If every", + "point of $S$ is an interior point", + "(that is,", + "$S^0=S$), then", + "$S$ is", + "{\\it open\\/}. A set $S$ is {\\it closed\\/} if", + "$S^c$ is open." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 371, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.12", + "categories": [], + "title": "", + "contents": [ + " Then", + "\\begin{alist}", + "\\item % (a)", + "$u_0$ is a {\\it limit point\\/} of $S$ if every deleted neighborhood of", + "$u_0$ contains a point of~$S$.", + "\\item % (b)", + "$u_0$ is a {\\it boundary", + "point\\/} of $S$ if every neighborhood of $u_0$ contains at least one point", + "in $S$ and one not in $S$. The set of boundary points of $S$ is the {\\it", + "boundary\\/} of $S$, denoted by $\\partial S$. The {\\it closure\\/} of $S$,", + "denoted by $\\overline{S}$, is defined by $\\overline{S}=S\\cup \\partial S$.", + "\\item % (c)", + "$u_0$ is an {\\it isolated point\\/} of $S$ if $u_0\\in S$ and there is a", + "neighborhood of $u_0$ that contains no other point of $S$.", + "\\item % (d)", + "$u_0$ is {\\it exterior } to $S$ if $u_0$ is in the interior of $S^c$. The", + "collection of such points is the {\\it exterior\\/} of $S$.", + "\\bbox", + "\\end{alist}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 372, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.14", + "categories": [], + "title": "", + "contents": [ + "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$", + " {\\it converges\\/}", + "to", + "$u\\in A$ if", + "\\begin{equation} \\label{eq:8.1.16}", + "\\lim_{n\\to\\infty}\\rho(u_n,u)=0.", + "\\end{equation}", + "In this case we say that", + "$\\lim_{n\\to\\infty}u_n=u$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 373, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.16", + "categories": [], + "title": "", + "contents": [ + "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is", + " a {\\it Cauchy sequence\\/}", + " if for every", + "$\\epsilon>0$ there is an integer $N$ such that", + "\\begin{equation} \\label{eq:8.1.17}", + "\\rho(u_n,u_m)<\\epsilon\\mbox{\\quad and \\quad}m,n>N.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 374, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.18", + "categories": [], + "title": "", + "contents": [ + "A metric space $(A,\\rho)$ is {\\it complete\\/}", + " if every Cauchy sequence in $A$", + "has a limit." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 375, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.1.20", + "categories": [], + "title": "", + "contents": [ + "If $\\rho$ and $\\sigma$ are both metrics on a set $A$, then $\\rho$", + "and $\\sigma$ are {\\it equivalent \\/}", + "\\hskip-.2em if there are positive constants $\\alpha$ and $\\beta$", + "such that", + "\\begin{equation} \\label{eq:8.1.18}", + "\\alpha\\le\\frac{\\rho(x,y)}{\\sigma(x,y)}\\le\\beta", + "\\mbox{\\quad for all \\quad}x,y\\in A\\mbox{\\quad such that \\quad}x\\ne y.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 376, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.1", + "categories": [], + "title": "", + "contents": [ + "The {\\it diameter\\/} of a nonempty subset $S$ of $A$ is", + "$$", + "d(S)=\\sup\\set{\\rho(u,v)}{u,\\, v\\in T}.", + "$$", + "If $d(S)<\\infty$ then $S$ is {\\it bounded\\/}.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 377, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.2", + "categories": [], + "title": "", + "contents": [ + "A set $T$ is {\\it compact\\/} if", + "it has the Heine--Borel property." + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 378, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.7", + "categories": [], + "title": "", + "contents": [ + "A set $T$ is {\\it totally bounded\\/}", + " if for every", + "$\\epsilon>0$", + "there is a finite set $T_\\epsilon$ with the following property:", + "if $t\\in T$, there is an $s\\in T_\\epsilon$ such that", + "$\\rho(s,t)<\\epsilon$.", + "We say that $T_\\epsilon$ is a {\\it finite $\\epsilon$-net for $T$\\/}.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 379, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.2.10", + "categories": [], + "title": "", + "contents": [ + "A subset $T$ of $C[a,b]$ is {\\it uniformly bounded\\/} if there is a", + "constant $M$ such that", + "\\begin{equation} \\label{eq:8.2.6}", + "|f(x)|\\le M \\mbox{\\quad if \\quad} a\\le x\\le b\\mbox{\\quad and \\quad}", + "f\\in T.", + "\\end{equation}", + "A subset $T$ of $C[a,b]$ is {\\it", + "equicontinuous\\/} if for each", + "$\\epsilon>0$ there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:8.2.7}", + "|f(x_1)-f(x_2)|\\le \\epsilon \\mbox{\\quad if \\quad}", + "x_1,x_2\\in [a,b],\\quad |x_1-x_2|<\\delta,\\mbox{\\quad and \\quad}f\\in T.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 380, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.1", + "categories": [], + "title": "", + "contents": [ + "We say that", + "$$", + "\\lim_{u\\to \\widehat u}f(u)=\\widehat v", + "$$", + "if $\\widehat u\\in\\overline D_f$ and for each $\\epsilon>0$ there is a", + "$\\delta>0$ such that", + "\\begin{equation} \\label{eq:8.3.1}", + "\\sigma(f(u),\\widehat v)<\\epsilon\\mbox{\\quad if \\quad}", + "u\\in D_f", + "\\mbox{\\quad and \\quad}", + "0<\\rho(u,\\widehat u)<\\delta.", + "\\end{equation}" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 381, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.2", + "categories": [], + "title": "", + "contents": [ + "We say that $f$", + "is {\\it continuous\\/} at", + "$\\widehat u$ if", + "$\\widehat u\\in D_f$ and for each $\\epsilon>0$", + "there is a $\\delta>0$ such that", + "\\begin{equation} \\label{eq:8.3.2}", + "\\sigma(f(u),f(\\widehat u))<\\epsilon\\mbox{\\quad if \\quad}", + "u\\in D_f\\cap N_\\delta(\\widehat u).", + "\\end{equation}", + "If $f$ is continuous at every point of a set $S$,", + "then $f$ is {\\it continuous on\\/} S.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 382, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.7", + "categories": [], + "title": "", + "contents": [ + "A function $f$ is {\\it uniformly continuous\\/} on a subset $S$ of $D_f$ if", + "for each $\\epsilon>0$ there is a $\\delta>0$ such that", + "$$", + "\\sigma(f(u),f(v))<\\epsilon\\mbox{\\quad whenever \\quad}", + "\\rho(u,v)<\\delta\\mbox{\\quad and \\quad}u,v\\in S.", + "$$" + ], + "refs": [], + "ref_ids": [] + }, + { + "id": 383, + "type": "definition", + "label": "TRENCH_REAL_ANALYSIS-thmtype:8.3.9", + "categories": [], + "title": "", + "contents": [ + "If $f:(A,\\rho)\\to (A,\\rho)$ is defined on all of $A$", + "and there is a constant $\\alpha$ in $(0,1)$", + "such that", + "\\begin{equation} \\label{eq:8.3.7}", + "\\rho(f(u),f(v))\\le\\alpha\\rho(u,v)", + "\\mbox{\\quad for all\\quad} (u,v)\\in A\\times A,", + "\\end{equation}", + "then $f$ is a {\\it contraction\\/} of $(A,\\rho)$.", + "\\bbox" + ], + "refs": [], + "ref_ids": [] + } + ], + "others": [], + "retrieval_examples": [ + 3, + 4, + 7, + 8, + 11, + 12, + 13, + 14, + 17, + 22, + 26, + 27, + 29, + 30, + 31, + 32, + 33, + 35, + 39, + 43, + 44, + 45, + 47, + 48, + 50, + 51, + 52, + 53, + 56, + 57, + 58, + 59, + 60, + 62, + 63, + 64, + 65, + 66, + 67, + 68, + 69, + 70, + 71, + 72, + 73, + 74, + 75, + 76, + 80, + 81, + 83, + 85, + 86, + 87, + 88, + 89, + 91, + 93, + 95, + 98, + 99, + 100, + 101, + 102, + 103, + 104, + 105, + 106, + 108, + 109, + 110, + 112, + 113, + 114, + 117, + 121, + 122, + 123, + 124, + 128, + 129, + 130, + 131, + 135, + 138, + 142, + 143, + 144, + 153, + 162, + 163, + 164, + 165, + 166, + 167, + 175, + 176, + 177, + 181, + 183, + 184, + 186, + 187, + 188, + 189, + 193, + 195, + 196, + 197, + 198, + 199, + 207, + 208, + 209, + 210, + 214, + 215, + 216, + 217, + 218, + 220, + 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