diff --git "a/naturalproofs_trench.json" "b/naturalproofs_trench.json" deleted file mode 100644--- "a/naturalproofs_trench.json" +++ /dev/null @@ -1,21023 +0,0 @@ -{ - "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": [], - 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