Normalize formatting further, macros for lean commands.
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@ -1,14 +1,10 @@
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\documentclass{report}
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\documentclass{report}
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\usepackage{graphicx}
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\usepackage{graphicx}
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\input{../preamble}
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\graphicspath{{./Apostol/images/}}
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\graphicspath{{./Apostol/images/}}
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\newcommand{\lean}[2]{\leanref{../#1.html\##2}{#2}}
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\input{../preamble}
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\newcommand{\leanPretty}[3]{\leanref{../#1.html\##2}{#3}}
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\makeleancommands{..}
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\newcommand{\ubar}[1]{\text{\b{$#1$}}}
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\begin{document}
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\begin{document}
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@ -228,7 +224,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\section{\verified{Lemma 1}}%
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\section{\verified{Lemma 1}}%
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\label{sec:lemma-1}
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\label{sec:lemma-1}
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\begin{lemma}{1}
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\begin{lemma}[1]
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Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
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Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
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@ -239,8 +235,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
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{Apostol.Chapter\_I\_03.is\_lub\_neg\_set\_iff\_is\_glb\_set\_neg}
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\divider
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Suppose $L = \sup{S}$ and fix $x \in S$.
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Suppose $L = \sup{S}$ and fix $x \in S$.
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By definition of the \nameref{sec:def-supremum}, $x \leq L$ and $L$ is the
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By definition of the \nameref{sec:def-supremum}, $x \leq L$ and $L$ is the
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smallest value satisfying this inequality.
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smallest value satisfying this inequality.
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@ -253,7 +247,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\section{\verified{Theorem I.27}}%
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\section{\verified{Theorem I.27}}%
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\label{sec:theorem-i.27}
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\label{sec:theorem-i.27}
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\begin{theorem}{I.27}
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\begin{theorem}[I.27]
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Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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is, there is a real number $L$ such that $L = \inf{S}$.
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is, there is a real number $L$ such that $L = \inf{S}$.
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@ -265,8 +259,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_isGLB}
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{Apostol.Chapter\_I\_03.exists\_isGLB}
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\divider
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Let $S$ be a nonempty set bounded below by $x$.
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Let $S$ be a nonempty set bounded below by $x$.
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Then $-S$ is nonempty and bounded above by $x$.
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Then $-S$ is nonempty and bounded above by $x$.
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By the \nameref{sec:completeness-axiom}, there exists a
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By the \nameref{sec:completeness-axiom}, there exists a
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@ -279,7 +271,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\section{\verified{Theorem I.29}}%
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\section{\verified{Theorem I.29}}%
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\label{sec:theorem-i.29}
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\label{sec:theorem-i.29}
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\begin{theorem}{I.29}
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\begin{theorem}[I.29]
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For every real $x$ there exists a positive integer $n$ such that $n > x$.
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For every real $x$ there exists a positive integer $n$ such that $n > x$.
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@ -290,8 +282,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_pnat\_geq\_self}
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{Apostol.Chapter\_I\_03.exists\_pnat\_geq\_self}
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\divider
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Let $n = \abs{\ceil{x}} + 1$.
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Let $n = \abs{\ceil{x}} + 1$.
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It is trivial to see $n$ is a positive integer satisfying $n \geq 1$.
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It is trivial to see $n$ is a positive integer satisfying $n \geq 1$.
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Thus all that remains to be shown is that $n > x$.
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Thus all that remains to be shown is that $n > x$.
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@ -305,7 +295,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\label{sec:archimedean-property-reals}
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\label{sec:archimedean-property-reals}
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\label{sec:theorem-i.30}
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\label{sec:theorem-i.30}
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\begin{theorem}{I.30}
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\begin{theorem}[I.30]
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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integer $n$ such that $nx > y$.
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integer $n$ such that $nx > y$.
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@ -317,8 +307,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.exists\_pnat\_mul\_self\_geq\_of\_pos}
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{Apostol.Chapter\_I\_03.exists\_pnat\_mul\_self\_geq\_of\_pos}
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\divider
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Let $x > 0$ and $y$ be an arbitrary real number.
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Let $x > 0$ and $y$ be an arbitrary real number.
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By \nameref{sec:theorem-i.29}, there exists a positive integer $n$ such that
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By \nameref{sec:theorem-i.29}, there exists a positive integer $n$ such that
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$n > y / x$.
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$n > y / x$.
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@ -329,7 +317,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\section{\verified{Theorem I.31}}%
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\section{\verified{Theorem I.31}}%
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\label{sec:theorem-i.31}
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\label{sec:theorem-i.31}
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\begin{theorem}{I.31}
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\begin{theorem}[I.31]
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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$$a \leq x \leq a + \frac{y}{n}$$ for every integer $n \geq 1$, then
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$$a \leq x \leq a + \frac{y}{n}$$ for every integer $n \geq 1$, then
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@ -342,8 +330,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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{Apostol.Chapter\_I\_03.forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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\divider
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By the trichotomy of the reals, there are three cases to consider:
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By the trichotomy of the reals, there are three cases to consider:
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\paragraph{Case 1}%
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\paragraph{Case 1}%
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@ -378,7 +364,7 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\section{\verified{Lemma 2}}%
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\section{\verified{Lemma 2}}%
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\label{sec:lemma-2}
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\label{sec:lemma-2}
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\begin{lemma}{2}
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\begin{lemma}[2]
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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$$a - y / n \leq x \leq a$$ for every integer $n \geq 1$, then $x = a$.
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$$a - y / n \leq x \leq a$$ for every integer $n \geq 1$, then $x = a$.
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@ -390,8 +376,6 @@ Every nonempty set $S$ of real numbers which is bounded above has a supremum;
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
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{Apostol.Chapter\_I\_03.forall\_pnat\_frac\_leq\_self\_leq\_imp\_eq}
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\divider
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By the trichotomy of the reals, there are three cases to consider:
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By the trichotomy of the reals, there are three cases to consider:
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\paragraph{Case 1}%
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\paragraph{Case 1}%
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@ -431,7 +415,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\subsection{\verified{Theorem I.32a}}%
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\subsection{\verified{Theorem I.32a}}%
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\label{sub:theorem-i.32a}
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\label{sub:theorem-i.32a}
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\begin{theorem}{I.32a}
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\begin{theorem}[I.32a]
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If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
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If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
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@ -442,8 +426,6 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.sup\_imp\_exists\_gt\_sup\_sub\_delta}
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{Apostol.Chapter\_I\_03.sup\_imp\_exists\_gt\_sup\_sub\_delta}
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\divider
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By definition of a \nameref{sec:def-supremum}, $\sup{S}$ is the least upper
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By definition of a \nameref{sec:def-supremum}, $\sup{S}$ is the least upper
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bound of $S$.
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bound of $S$.
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For the sake of contradiction, suppose for all $x \in S$,
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For the sake of contradiction, suppose for all $x \in S$,
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@ -459,7 +441,7 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\subsection{\verified{Theorem I.32b}}%
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\subsection{\verified{Theorem I.32b}}%
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\label{sub:theorem-i.32b}
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\label{sub:theorem-i.32b}
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\begin{theorem}{I.32b}
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\begin{theorem}[I.32b]
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If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
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If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
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@ -470,8 +452,6 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.inf\_imp\_exists\_lt\_inf\_add\_delta}
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{Apostol.Chapter\_I\_03.inf\_imp\_exists\_lt\_inf\_add\_delta}
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\divider
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By definition of an \nameref{sec:def-infimum}, $\inf{S}$ is the greatest lower
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By definition of an \nameref{sec:def-infimum}, $\inf{S}$ is the greatest lower
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bound of $S$.
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bound of $S$.
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For the sake of contradiction, suppose for all $x \in S$,
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For the sake of contradiction, suppose for all $x \in S$,
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\subsection{\verified{Theorem I.33a}}%
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\subsection{\verified{Theorem I.33a}}%
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\label{sub:theorem-i.33a}
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\label{sub:theorem-i.33a}
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\begin{theorem}{I.33a}
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\begin{theorem}[I.33a]
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If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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$$\sup{C} = \sup{A} + \sup{B}.$$
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$$\sup{C} = \sup{A} + \sup{B}.$$
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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{Apostol.Chapter\_I\_03.sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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\divider
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We prove (i) $\sup{A} + \sup{B}$ is an upper bound of $C$ and (ii)
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We prove (i) $\sup{A} + \sup{B}$ is an upper bound of $C$ and (ii)
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$\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
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$\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
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\subsection{\verified{Theorem I.33b}}%
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\subsection{\verified{Theorem I.33b}}%
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\label{sub:theorem-i.33b}
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\label{sub:theorem-i.33b}
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\begin{theorem}{I.33b}
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\begin{theorem}[I.33b]
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If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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$$\inf{C} = \inf{A} + \inf{B}.$$
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$$\inf{C} = \inf{A} + \inf{B}.$$
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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{Apostol.Chapter\_I\_03.inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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\divider
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We prove (i) $\inf{A} + \inf{B}$ is a lower bound of $C$ and (ii)
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We prove (i) $\inf{A} + \inf{B}$ is a lower bound of $C$ and (ii)
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$\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
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$\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
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\section{\verified{Theorem I.34}}%
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\section{\verified{Theorem I.34}}%
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\label{sec:theorem-i.34}
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\label{sec:theorem-i.34}
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\begin{theorem}{I.34}
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\begin{theorem}[I.34]
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Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
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Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
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for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
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for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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\lean{Bookshelf/Apostol/Chapter\_I\_03}
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{Apostol.Chapter\_I\_03.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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{Apostol.Chapter\_I\_03.forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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\divider
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By hypothesis, $S$ and $T$ are nonempty sets.
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By hypothesis, $S$ and $T$ are nonempty sets.
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Let $s \in S$ and $t \in T$.
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Let $s \in S$ and $t \in T$.
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Then $t$ is an upper bound of $S$ and $s$ is a lower bound of $T$.
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Then $t$ is an upper bound of $S$ and $s$ is a lower bound of $T$.
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\begin{axiom}
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\begin{axiom}
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\leanPretty{Common/Geometry/Area}{Nonnegative-Property}
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\leanp{Common/Geometry/Area}{Nonnegative-Property}
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{Nonnegative Property}
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{Nonnegative Property}
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\end{axiom}
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\end{axiom}
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\begin{axiom}
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\begin{axiom}
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\leanPretty{Common/Geometry/Area}{Additive-Property}
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\leanp{Common/Geometry/Area}{Additive-Property}
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{Additive Property}
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{Additive Property}
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\end{axiom}
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\end{axiom}
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@ -715,7 +689,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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\begin{axiom}
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\begin{axiom}
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\leanPretty{Common/Geometry/Area}{Difference-Property}
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\leanp{Common/Geometry/Area}{Difference-Property}
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{Difference Property}
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{Difference Property}
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\end{axiom}
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\end{axiom}
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\begin{axiom}
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\begin{axiom}
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\leanPretty{Common/Geometry/Area}{Invariance-Under-Congruence}
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\leanp{Common/Geometry/Area}{Invariance-Under-Congruence}
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{Invariance Under Congruence}
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{Invariance Under Congruence}
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\end{axiom}
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\end{axiom}
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\begin{axiom}
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\begin{axiom}
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\leanPretty{Common/Geometry/Area}{Choice-of-Scale}
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\leanp{Common/Geometry/Area}{Choice-of-Scale}
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{Choice of Scale}
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{Choice of Scale}
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\end{axiom}
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\end{axiom}
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|
||||||
\begin{axiom}
|
\begin{axiom}
|
||||||
|
|
||||||
\leanPretty{Common/Geometry/Area}{Exhaustion-Property}
|
\leanp{Common/Geometry/Area}{Exhaustion-Property}
|
||||||
{Exhaustion Property}
|
{Exhaustion Property}
|
||||||
|
|
||||||
\end{axiom}
|
\end{axiom}
|
||||||
|
@ -1045,7 +1019,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
|
||||||
These cases are exhaustive and in agreement with one another.
|
These cases are exhaustive and in agreement with one another.
|
||||||
Thus $S$ is measurable and $$a(S) = \frac{b_1 + b_2}{2}h.$$
|
Thus $S$ is measurable and $$a(S) = \frac{b_1 + b_2}{2}h.$$
|
||||||
|
|
||||||
\divider
|
\suitdivider
|
||||||
|
|
||||||
Let $P$ be a parallelogram with base $b$ and height $h$.
|
Let $P$ be a parallelogram with base $b$ and height $h$.
|
||||||
Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
|
Then $P$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
|
||||||
|
@ -1355,8 +1329,6 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4a}
|
{Apostol.Chapter\_1\_11.exercise\_4a}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
Let $x$ be a real number and $n$ an integer.
|
Let $x$ be a real number and $n$ an integer.
|
||||||
Let $m = \floor{x + n}$.
|
Let $m = \floor{x + n}$.
|
||||||
By definition of the floor function, $m$ is the unique integer such that
|
By definition of the floor function, $m$ is the unique integer such that
|
||||||
|
@ -1380,14 +1352,12 @@ $\floor{-x} =
|
||||||
|
|
||||||
\ \vspace{6pt}
|
\ \vspace{6pt}
|
||||||
|
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean*{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4b\_1}
|
{Apostol.Chapter\_1\_11.exercise\_4b\_1}
|
||||||
|
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4b\_2}
|
{Apostol.Chapter\_1\_11.exercise\_4b\_2}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
There are two cases to consider:
|
There are two cases to consider:
|
||||||
|
|
||||||
\paragraph{Case 1}%
|
\paragraph{Case 1}%
|
||||||
|
@ -1428,8 +1398,6 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4c}
|
{Apostol.Chapter\_1\_11.exercise\_4c}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
Rewrite $x$ and $y$ as the sum of their floor and fractional components:
|
Rewrite $x$ and $y$ as the sum of their floor and fractional components:
|
||||||
$x = \floor{x} + \{x\}$ and $y = \floor{y} + \{y\}$.
|
$x = \floor{x} + \{x\}$ and $y = \floor{y} + \{y\}$.
|
||||||
Now
|
Now
|
||||||
|
@ -1474,8 +1442,6 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4d}
|
{Apostol.Chapter\_1\_11.exercise\_4d}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
This is immediately proven by applying \nameref{sec:hermites-identity}.
|
This is immediately proven by applying \nameref{sec:hermites-identity}.
|
||||||
|
|
||||||
\end{proof}
|
\end{proof}
|
||||||
|
@ -1490,8 +1456,6 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_4e}
|
{Apostol.Chapter\_1\_11.exercise\_4e}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
This is immediately proven by applying \nameref{sec:hermites-identity}.
|
This is immediately proven by applying \nameref{sec:hermites-identity}.
|
||||||
|
|
||||||
\end{proof}
|
\end{proof}
|
||||||
|
@ -1508,8 +1472,6 @@ State and prove such a generalization.
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_5}
|
{Apostol.Chapter\_1\_11.exercise\_5}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
We prove that for all natural numbers $n$ and real numbers $x$, the following
|
We prove that for all natural numbers $n$ and real numbers $x$, the following
|
||||||
identity holds:
|
identity holds:
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
|
@ -1734,8 +1696,6 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
|
||||||
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
\lean{Bookshelf/Apostol/Chapter\_1\_11}
|
||||||
{Apostol.Chapter\_1\_11.exercise\_7b}
|
{Apostol.Chapter\_1\_11.exercise\_7b}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
Let $n = 1, \ldots, b - 1$.
|
Let $n = 1, \ldots, b - 1$.
|
||||||
By hypothesis, $a$ and $b$ are coprime.
|
By hypothesis, $a$ and $b$ are coprime.
|
||||||
Furthermore, $n < b$ for all values of $n$.
|
Furthermore, $n < b$ for all values of $n$.
|
||||||
|
@ -1807,7 +1767,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-additive-property}
|
\label{sec:step-additive-property}
|
||||||
\label{sec:theorem-1.2}
|
\label{sec:theorem-1.2}
|
||||||
|
|
||||||
\begin{theorem}{1.2}
|
\begin{theorem}[1.2]
|
||||||
|
|
||||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
||||||
$[a, b]$.
|
$[a, b]$.
|
||||||
|
@ -1850,7 +1810,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-homogeneous-property}
|
\label{sec:step-homogeneous-property}
|
||||||
\label{sec:theorem-1.3}
|
\label{sec:theorem-1.3}
|
||||||
|
|
||||||
\begin{theorem}{1.3}
|
\begin{theorem}[1.3]
|
||||||
|
|
||||||
Let $s$ be a \nameref{sec:def-step-function} on closed interval $[a, b]$.
|
Let $s$ be a \nameref{sec:def-step-function} on closed interval $[a, b]$.
|
||||||
For every real number $c$, we have
|
For every real number $c$, we have
|
||||||
|
@ -1881,7 +1841,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-linearity-property}
|
\label{sec:step-linearity-property}
|
||||||
\label{sec:theorem-1.4}
|
\label{sec:theorem-1.4}
|
||||||
|
|
||||||
\begin{theorem}{1.4}
|
\begin{theorem}[1.4]
|
||||||
|
|
||||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
||||||
$[a, b]$.
|
$[a, b]$.
|
||||||
|
@ -1911,7 +1871,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-comparison-theorem}
|
\label{sec:step-comparison-theorem}
|
||||||
\label{sec:theorem-1.5}
|
\label{sec:theorem-1.5}
|
||||||
|
|
||||||
\begin{theorem}{1.5}
|
\begin{theorem}[1.5]
|
||||||
|
|
||||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval
|
||||||
$[a, b]$.
|
$[a, b]$.
|
||||||
|
@ -1950,7 +1910,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-additivity-with-respect-interval-integration}
|
\label{sec:step-additivity-with-respect-interval-integration}
|
||||||
\label{sec:theorem-1.6}
|
\label{sec:theorem-1.6}
|
||||||
|
|
||||||
\begin{theorem}{1.6}
|
\begin{theorem}[1.6]
|
||||||
|
|
||||||
Let $a, b, c \in \mathbb{R}$ and $s$ a \nameref{sec:def-step-function} on the
|
Let $a, b, c \in \mathbb{R}$ and $s$ a \nameref{sec:def-step-function} on the
|
||||||
smallest closed interval containing them.
|
smallest closed interval containing them.
|
||||||
|
@ -1996,7 +1956,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-invariance-under-translation}
|
\label{sec:step-invariance-under-translation}
|
||||||
\label{sec:theorem-1.7}
|
\label{sec:theorem-1.7}
|
||||||
|
|
||||||
\begin{theorem}{1.7}
|
\begin{theorem}[1.7]
|
||||||
|
|
||||||
Let $s$ be a step function on closed interval $[a, b]$.
|
Let $s$ be a step function on closed interval $[a, b]$.
|
||||||
Then
|
Then
|
||||||
|
@ -2037,7 +1997,7 @@ This property is described by saying that every step function is a linear
|
||||||
\label{sec:step-expansion-contraction-interval-integration}
|
\label{sec:step-expansion-contraction-interval-integration}
|
||||||
\label{sec:theorem-1.8}
|
\label{sec:theorem-1.8}
|
||||||
|
|
||||||
\begin{theorem}{1.8}
|
\begin{theorem}[1.8]
|
||||||
|
|
||||||
Let $s$ be a step function on closed interval $[a, b]$.
|
Let $s$ be a step function on closed interval $[a, b]$.
|
||||||
Then
|
Then
|
||||||
|
@ -2425,13 +2385,13 @@ Which of the following properties would remain valid in this new theory?
|
||||||
$\int_a^b s + \int_b^c s = \int_a^c s$.
|
$\int_a^b s + \int_b^c s = \int_a^c s$.
|
||||||
|
|
||||||
\note{This property mirrors
|
\note{This property mirrors
|
||||||
\nameref{sec:step-additivity-with-respect-interval-integration}}
|
\nameref{sec:step-additivity-with-respect-interval-integration}.}
|
||||||
|
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
|
||||||
The above property is valid.
|
The above property is \textbf{valid}.
|
||||||
|
|
||||||
\divider
|
\vspace{6pt}
|
||||||
|
|
||||||
WLOG, suppose $a < b < c$.
|
WLOG, suppose $a < b < c$.
|
||||||
Let $s$ be a step function defined on closed interval $[a, c]$.
|
Let $s$ be a step function defined on closed interval $[a, c]$.
|
||||||
|
@ -2464,9 +2424,9 @@ $\int_a^b (s + t) = \int_a^b s + \int_a^b t$.
|
||||||
|
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
|
||||||
The above property is invalid.
|
The above property is \textbf{invalid}.
|
||||||
|
|
||||||
\divider
|
\vspace{6pt}
|
||||||
|
|
||||||
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
||||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||||
|
@ -2511,9 +2471,9 @@ $\int_a^b c \cdot s = c \int_a^b s$.
|
||||||
|
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
|
||||||
The above property is invalid.
|
The above property is \textbf{invalid}.
|
||||||
|
|
||||||
\divider
|
\vspace{6pt}
|
||||||
|
|
||||||
Let $s$ be a step function on closed interval $[a, b]$.
|
Let $s$ be a step function on closed interval $[a, b]$.
|
||||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||||
|
@ -2543,9 +2503,9 @@ $\int_{a+c}^{b+c} s(x) \mathop{dx} = \int_a^b s(x + c) \mathop{dx}$.
|
||||||
|
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
|
||||||
The above property is valid.
|
The above property is \textbf{valid}.
|
||||||
|
|
||||||
\divider
|
\vspace{6pt}
|
||||||
|
|
||||||
Let $s$ be a step function on closed interval $[a + c, b + c]$.
|
Let $s$ be a step function on closed interval $[a + c, b + c]$.
|
||||||
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$.
|
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition} $P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open subinterval of $P$.
|
||||||
|
@ -2577,9 +2537,9 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
|
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
|
||||||
The above property is valid.
|
The above property is \textbf{valid}.
|
||||||
|
|
||||||
\divider
|
\vspace{6pt}
|
||||||
|
|
||||||
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
||||||
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition}
|
By definition of a \nameref{sec:def-step-function}, there exists a \nameref{sec:def-partition}
|
||||||
|
@ -2611,7 +2571,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
\section{\partial{Theorem 1.9}}%
|
\section{\partial{Theorem 1.9}}%
|
||||||
\label{sec:theorem-1.9}
|
\label{sec:theorem-1.9}
|
||||||
|
|
||||||
\begin{theorem}{1.9}
|
\begin{theorem}[1.9]
|
||||||
|
|
||||||
Every function $f$ which is bounded on $[a, b]$ has a lower integral
|
Every function $f$ which is bounded on $[a, b]$ has a lower integral
|
||||||
$\ubar{I}(f)$ and an upper integral $\overline{I}(f)$ satisfying the
|
$\ubar{I}(f)$ and an upper integral $\overline{I}(f)$ satisfying the
|
||||||
|
@ -2677,7 +2637,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
\label{sec:measurability-ordinate-sets}
|
\label{sec:measurability-ordinate-sets}
|
||||||
\label{sec:theorem-1.10}
|
\label{sec:theorem-1.10}
|
||||||
|
|
||||||
\begin{theorem}{1.10}
|
\begin{theorem}[1.10]
|
||||||
|
|
||||||
Let $f$ be a nonnegative function, \nameref{sec:def-integrable} on an interval
|
Let $f$ be a nonnegative function, \nameref{sec:def-integrable} on an interval
|
||||||
$[a, b]$, and let $Q$ denote the ordinate set of $f$ over $[a, b]$.
|
$[a, b]$, and let $Q$ denote the ordinate set of $f$ over $[a, b]$.
|
||||||
|
@ -2705,7 +2665,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
\label{sec:measurability-graph-nonnegative-function}
|
\label{sec:measurability-graph-nonnegative-function}
|
||||||
\label{sec:theorem-1.11}
|
\label{sec:theorem-1.11}
|
||||||
|
|
||||||
\begin{theorem}{1.11}
|
\begin{theorem}[1.11]
|
||||||
|
|
||||||
Let $f$ be a nonnegative function, integrable on an interval $[a, b]$.
|
Let $f$ be a nonnegative function, integrable on an interval $[a, b]$.
|
||||||
Then the graph of $f$, that is, the set
|
Then the graph of $f$, that is, the set
|
||||||
|
@ -2761,7 +2721,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
\label{sec:integrability-bounded-monotonic-functions}
|
\label{sec:integrability-bounded-monotonic-functions}
|
||||||
\label{sec:theorem-1.12}
|
\label{sec:theorem-1.12}
|
||||||
|
|
||||||
\begin{theorem}{1.12}
|
\begin{theorem}[1.12]
|
||||||
|
|
||||||
If $f$ is \nameref{sec:def-monotonic} on a closed interval $[a, b]$, then $f$
|
If $f$ is \nameref{sec:def-monotonic} on a closed interval $[a, b]$, then $f$
|
||||||
is \nameref{sec:def-integrable} on $[a, b]$.
|
is \nameref{sec:def-integrable} on $[a, b]$.
|
||||||
|
@ -2867,7 +2827,7 @@ If $s(x) < t(x)$ for each $x$ in $[a, b]$, then $\int_a^b s < \int_a^b t$.
|
||||||
\label{sec:calculation-integral-bounded-monotonic-function}
|
\label{sec:calculation-integral-bounded-monotonic-function}
|
||||||
\label{sec:theorem-1.13}
|
\label{sec:theorem-1.13}
|
||||||
|
|
||||||
\begin{theorem}{1.13}
|
\begin{theorem}[1.13]
|
||||||
|
|
||||||
Assume $f$ is increasing on a closed interval $[a, b]$.
|
Assume $f$ is increasing on a closed interval $[a, b]$.
|
||||||
Let $x_k = a + k(b - a) / n$ for $k = 0, 1, \ldots, n$.
|
Let $x_k = a + k(b - a) / n$ for $k = 0, 1, \ldots, n$.
|
||||||
|
|
|
@ -1,9 +1,7 @@
|
||||||
\documentclass{report}
|
\documentclass{report}
|
||||||
|
|
||||||
\input{../preamble}
|
\input{../preamble}
|
||||||
|
\makeleancommands
|
||||||
\newcommand{\lean}[2]{\leanref{../#1.html\##2}{#2}}
|
|
||||||
\newcommand{\leanPretty}[3]{\leanref{../#1.html\##2}{#3}}
|
|
||||||
|
|
||||||
\begin{document}
|
\begin{document}
|
||||||
|
|
||||||
|
|
|
@ -1,8 +1,7 @@
|
||||||
\documentclass{article}
|
\documentclass{article}
|
||||||
|
|
||||||
\input{../../preamble}
|
\input{../../preamble}
|
||||||
|
\makeleancommands{../..}
|
||||||
\newcommand{\lean}[2]{\leanref{../../#1.html\##2}{#2}}
|
|
||||||
|
|
||||||
\begin{document}
|
\begin{document}
|
||||||
|
|
||||||
|
@ -28,8 +27,6 @@ Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
|
||||||
\lean{Common/Real/Sequence/Arithmetic}
|
\lean{Common/Real/Sequence/Arithmetic}
|
||||||
{Real.Arithmetic.sum\_recursive\_closed}
|
{Real.Arithmetic.sum\_recursive\_closed}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
|
Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
|
||||||
By definition, for all $k \in \mathbb{N}$,
|
By definition, for all $k \in \mathbb{N}$,
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
|
@ -90,8 +87,6 @@ Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
|
||||||
\lean{Common/Real/Sequence/Geometric}
|
\lean{Common/Real/Sequence/Geometric}
|
||||||
{Real.Geometric.sum\_recursive\_closed}
|
{Real.Geometric.sum\_recursive\_closed}
|
||||||
|
|
||||||
\divider
|
|
||||||
|
|
||||||
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
|
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
|
||||||
By definition, for all $k \in \mathbb{N}$,
|
By definition, for all $k \in \mathbb{N}$,
|
||||||
\begin{equation}
|
\begin{equation}
|
||||||
|
|
97
preamble.tex
97
preamble.tex
|
@ -1,6 +1,8 @@
|
||||||
\usepackage{amsfonts, amsmath, amsthm}
|
\usepackage{amsfonts, amsmath, amsthm}
|
||||||
|
\usepackage{bigfoot}
|
||||||
\usepackage{comment}
|
\usepackage{comment}
|
||||||
\usepackage[shortlabels]{enumitem}
|
\usepackage[shortlabels]{enumitem}
|
||||||
|
\usepackage{etoolbox}
|
||||||
\usepackage{environ}
|
\usepackage{environ}
|
||||||
\usepackage{fancybox}
|
\usepackage{fancybox}
|
||||||
\usepackage{fontawesome5}
|
\usepackage{fontawesome5}
|
||||||
|
@ -11,23 +13,63 @@
|
||||||
\usepackage{xr-hyper}
|
\usepackage{xr-hyper}
|
||||||
\usepackage{hyperref}
|
\usepackage{hyperref}
|
||||||
|
|
||||||
|
% Open "private" namespace.
|
||||||
|
\makeatletter
|
||||||
|
|
||||||
|
% ========================================
|
||||||
|
% General
|
||||||
|
% ========================================
|
||||||
|
|
||||||
|
\newcommand{\header}[2]{\title{#1}\author{#2}\date{}\maketitle}
|
||||||
|
|
||||||
|
% ========================================
|
||||||
|
% Dividers
|
||||||
|
% ========================================
|
||||||
|
|
||||||
|
\newcommand\@linespace{\vspace{10pt}}
|
||||||
|
\newcommand\linedivider{\@linespace\hrule\@linespace}
|
||||||
|
\WithSuffix\newcommand\linedivider*{\@linespace\hrule}
|
||||||
|
\newcommand\suitdivider{$$\spadesuit\spadesuit\spadesuit$$}
|
||||||
|
|
||||||
% ========================================
|
% ========================================
|
||||||
% Linking
|
% Linking
|
||||||
% ========================================
|
% ========================================
|
||||||
|
|
||||||
\hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue}
|
\hypersetup{colorlinks=true, linkcolor=blue, urlcolor=blue}
|
||||||
\newcommand{\leanref}[2]{\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1}{#2}}
|
|
||||||
\newcommand{\textref}[1]{\text{\nameref{#1}}}
|
\newcommand{\textref}[1]{\text{\nameref{#1}}}
|
||||||
|
|
||||||
|
\newcommand\@leanlink[4]{%
|
||||||
|
\textcolor{blue}{$\pmb{\exists}\;{-}\;$}\href{#1/#2.html\##3}{#4}}
|
||||||
|
|
||||||
|
% Reference to an anchor of Lean documentation.
|
||||||
|
\newcommand\leanref[3]{%
|
||||||
|
\@leanlink{#1}{#2}{#3}{#3}\vspace{10pt}}
|
||||||
|
\WithSuffix\newcommand\leanref*[3]{%
|
||||||
|
\@leanlink{#1}{#2}{#3}{#3}}
|
||||||
|
|
||||||
|
% Variant that allows customizing display text.
|
||||||
|
\newcommand\leanpref[4]{%
|
||||||
|
\@leanlink{#1}{#2}{#3}{#4}\vspace{10pt}}
|
||||||
|
\WithSuffix\newcommand\leanpref*[4]{%
|
||||||
|
\@leanlink{#1}{#2}{#3}{#4}}
|
||||||
|
|
||||||
|
% Macro to build all Lean related commands relative to a specified directory.
|
||||||
|
\newcommand\makeleancommands[1]{%
|
||||||
|
\newcommand\lean[2]{%
|
||||||
|
\leanref{#1}{##1}{##2}}
|
||||||
|
\WithSuffix\newcommand\lean*[2]{%
|
||||||
|
\leanref*{#1}{##1}{##2}}
|
||||||
|
\newcommand\leanp[3]{%
|
||||||
|
\leanpref{#1}{##1}{##2}{##3}}
|
||||||
|
\WithSuffix\newcommand\leanp*[3]{%
|
||||||
|
\leanpref*{#1}{##1}{##2}{##3}}
|
||||||
|
}
|
||||||
|
|
||||||
% ========================================
|
% ========================================
|
||||||
% Environments
|
% Admonitions
|
||||||
% ========================================
|
% ========================================
|
||||||
|
|
||||||
\newcommand{\divider}{\vspace{10pt}\hrule\vspace{10pt}}
|
\newcommand{\@admonition}[2]{%
|
||||||
\newcommand{\header}[2]{\title{#1}\author{#2}\date{}\maketitle}
|
|
||||||
|
|
||||||
% Admonitions.
|
|
||||||
\newcommand{\admonition}[2]{%
|
|
||||||
\begin{center}
|
\begin{center}
|
||||||
\doublebox{
|
\doublebox{
|
||||||
\begin{minipage}{0.95\textwidth}
|
\begin{minipage}{0.95\textwidth}
|
||||||
|
@ -36,26 +78,33 @@
|
||||||
\vspace{2pt}
|
\vspace{2pt}
|
||||||
\end{minipage}}
|
\end{minipage}}
|
||||||
\end{center}}
|
\end{center}}
|
||||||
\newcommand{\note}[1]{\admonition{Note:}{#1}}
|
|
||||||
\newcommand{\todo}[1]{\admonition{TODO:}{#1}}
|
|
||||||
|
|
||||||
% Statements.
|
\newcommand{\note}[1]{\@admonition{Note:}{#1}}
|
||||||
\newenvironment{axiom}{%
|
\newcommand{\todo}[1]{\@admonition{TODO:}{#1}}
|
||||||
\paragraph{\normalfont\normalsize\textit{Axiom.}}}
|
|
||||||
{\hfill$\square$}
|
% ========================================
|
||||||
\newenvironment{definition}{%
|
% Statements
|
||||||
\paragraph{\normalfont\normalsize\textit{Definition.}}}
|
% ========================================
|
||||||
{\hfill$\square$}
|
|
||||||
|
\newcommand\@statement[1]{%
|
||||||
|
\linedivider*\paragraph{\normalfont\normalsize\textit{#1.}}}
|
||||||
|
|
||||||
|
\newenvironment{axiom}{\@statement{Axiom}}{\hfill$\square$}
|
||||||
|
\newenvironment{definition}{\@statement{Definition}}{\hfill$\square$}
|
||||||
|
\renewenvironment{proof}{\@statement{Proof}}{\hfill$\square$}
|
||||||
|
|
||||||
\newtheorem{lemmainner}{Lemma}
|
\newtheorem{lemmainner}{Lemma}
|
||||||
\newenvironment{lemma}[1]{%
|
\newenvironment{lemma}[1][]{%
|
||||||
\renewcommand\thelemmainner{#1}%
|
\ifstrempty{#1}
|
||||||
\lemmainner
|
{\lemmainner}
|
||||||
|
{\renewcommand\thelemmainner{#1}\lemmainner}
|
||||||
}{\endlemmainner}
|
}{\endlemmainner}
|
||||||
|
|
||||||
\newtheorem{theoreminner}{Theorem}
|
\newtheorem{theoreminner}{Theorem}
|
||||||
\newenvironment{theorem}[1]{%
|
\newenvironment{theorem}[1][]{%
|
||||||
\renewcommand\thetheoreminner{#1}%
|
\ifstrempty{#1}
|
||||||
\theoreminner
|
{\theoreminner}
|
||||||
|
{\renewcommand\thetheoreminner{#1}\theoreminner}
|
||||||
}{\endtheoreminner}
|
}{\endtheoreminner}
|
||||||
|
|
||||||
% ========================================
|
% ========================================
|
||||||
|
@ -82,3 +131,7 @@
|
||||||
\newcommand{\ico}[2]{\left[#1, #2\right)}
|
\newcommand{\ico}[2]{\left[#1, #2\right)}
|
||||||
\newcommand{\ioc}[2]{\left(#1, #2\right]}
|
\newcommand{\ioc}[2]{\left(#1, #2\right]}
|
||||||
\newcommand{\ioo}[2]{\left(#1, #2\right)}
|
\newcommand{\ioo}[2]{\left(#1, #2\right)}
|
||||||
|
\newcommand{\ubar}[1]{\text{\b{$#1$}}}
|
||||||
|
|
||||||
|
% Close off "private" namespace.
|
||||||
|
\makeatother
|
||||||
|
|
Loading…
Reference in New Issue