Start working on Apostol exercises 1.7.
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@ -2,8 +2,11 @@
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\input{preamble}
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\newcommand{\link}[1]{\lean{../../..}{Bookshelf/Real/Sequence/Arithmetic}
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{Real.Arithmetic.#1}}
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\newcommand{\link}[1]{\lean{../../..}
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{Bookshelf/Real/Sequence/Arithmetic}
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{Real.Arithmetic.#1}
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{Real.Arithmetic.#1}
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}
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\begin{document}
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@ -16,7 +19,7 @@ $$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
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\begin{proof}
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\link{sum_recursive_closed}
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\link{sum\_recursive\_closed}
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\end{proof}
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@ -2,8 +2,11 @@
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\input{preamble}
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\newcommand{\link}[1]{\lean{../../..}{Bookshelf/Real/Sequence/Geometric}
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{Real.Geometric.#1}}
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\newcommand{\link}[1]{\lean{../../..}
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{Bookshelf/Real/Sequence/Geometric}
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{Real.Geometric.#1}
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{Real.Geometric.#1}
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}
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\begin{document}
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@ -16,7 +19,7 @@ $$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
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\begin{proof}
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\link{sum_recursive_closed}
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\link{sum\_recursive\_closed}
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\end{proof}
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@ -3,8 +3,11 @@
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\input{preamble}
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\newcommand{\link}[1]{\lean{../..}{Exercises/Apostol/Chapter_I_3}
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{Exercises.Apostol.Chapter\_I\_3.Real.#1}}
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\newcommand{\link}[1]{\lean{../..}
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{Exercises/Apostol/Chapter\_I\_3}
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{Exercises.Apostol.Chapter\_I\_3.Real.#1}
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{Chapter\_I\_3.Real.#1}
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}
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\begin{document}
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@ -16,7 +19,7 @@ is, there is a real number $L$ such that $L = \inf{S}$.
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\begin{proof}
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\link{exists_isGLB}
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\link{exists\_isGLB}
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\end{proof}
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@ -27,7 +30,7 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\begin{proof}
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\link{exists_pnat_geq_self}
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\link{exists\_pnat\_geq\_self}
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\end{proof}
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@ -39,7 +42,7 @@ integer $n$ such that $nx > y$.
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\begin{proof}
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\link{exists_pnat_mul_self_geq_of_pos}
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\link{exists\_pnat\_mul\_self\_geq\_of\_pos}
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\end{proof}
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@ -52,7 +55,7 @@ for every integer $n \geq 1$, then $x = a$.
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\begin{proof}
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\link{forall_pnat_leq_self_leq_frac_imp_eq}
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\link{forall\_pnat\_leq\_self\_leq\_frac\_imp\_eq}
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\end{proof}
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@ -70,8 +73,8 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
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\begin{proof}
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\begin{enumerate}[(a)]
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\item \link{sup_imp_exists_gt_sup_sub_delta}
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\item \link{inf_imp_exists_lt_inf_add_delta}
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\item \link{sup\_imp\_exists\_gt\_sup\_sub\_delta}
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\item \link{inf\_imp\_exists\_lt\_inf\_add\_delta}
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\end{enumerate}
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\end{proof}
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@ -92,8 +95,8 @@ $$C = \{a + b : a \in A, b \in B\}.$$
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\begin{proof}
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\begin{enumerate}[(a)]
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\item \link{sup_minkowski_sum_eq_sup_add_sup}
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\item \link{inf_minkowski_sum_eq_inf_add_inf}
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\item \link{sup\_minkowski\_sum\_eq\_sup\_add\_sup}
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\item \link{inf\_minkowski\_sum\_eq\_inf\_add\_inf}
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\end{enumerate}
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\end{proof}
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@ -109,7 +112,7 @@ $$\sup{S} \leq \inf{T}.$$
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\begin{proof}
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\link{forall_mem_le_forall_mem_imp_sup_le_inf}
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\link{forall\_mem\_le\_forall\_mem\_imp\_sup\_le\_inf}
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\end{proof}
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@ -0,0 +1,177 @@
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\documentclass{article}
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\usepackage{amsmath}
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\input{preamble}
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\newcommand{\larea}[2]{\lean{../..}{Bookshelf/Real/Geometry/Area}{#1}{#2}}
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\newcommand{\lrect}[2]{\lean{../..}{Bookshelf/Real/Geometry/Rectangle}{#1}{#2}}
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\begin{document}
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The properties of area in this set of exercises are to be deduced from the
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axioms for area stated in the foregoing section.
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\section{Exercise 1}%
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\label{sec:exercise-1}
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Prove that each of the following sets is measurable and has zero area:
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\subsection{Exercise 1a}%
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\label{sub:exercise-1a}
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A set consisting of a single point.
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\begin{proof}
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Let $S$ be a set consisting of a single point.
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By definition of a \lrect{Real.Point}{Point}, $S$ is a rectangle in which all
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vertices coincide.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width times its height.
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The width and height of $S$ is trivially zero.
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Therefore $a(S) = (0)(0) = 0$.
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\end{proof}
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\subsection{Exercise 1b}%
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\label{sub:exercise-1b}
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A set consisting of a finite number of points in a plane.
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\begin{proof}
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We show for all $k > 0$, a set consisting of $k$ points in a plane is
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measurable with area $0$.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single point in a plane.
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By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$.
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\paragraph{Induction Step}%
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Define our induction hypothesis as, "Let $k > 0$ and assume a set consisting
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of $k$ points in a plane is measurable with area $0$."
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Consider a set $S_{k+1}$ consisting of $k + 1$ points in a plane.
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Pick an arbitrary point of $S_{k+1}$.
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Denote the set containing just this point as $T$.
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Denote the remaining set of points as $S_k$.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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& = a(S_k \cup T) \nonumber \\
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& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
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& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1b-eq1}
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\end{align}
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\noindent
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There are two cases to consider:
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\subparagraph{Case 1}%
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$S_k \cap T = \emptyset$.
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Then it trivially follows that $a(S_k \cap T) = 0$.
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\subparagraph{Case 2}%
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$S_k \cap T \neq \emptyset$.
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Since $T$ consists of a single point, $S_k \cap T = T$.
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By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
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\vspace{8pt}
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\noindent
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In both cases, \eqref{sub:exercise-1b-eq1} evaluates to $0$, implying
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$a(S_{k+1}) = 0$ as expected.
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\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, a set consisting of
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$n$ points in a plane is measurable with area $0$.
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\end{proof}
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\subsection{Exercise 1c}%
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\label{sub:exercise-1c}
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The union of a finite collection of line segments in a plane.
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\begin{proof}
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We show for all $k > 0$, a set consisting of $k$ line segments in a plane is
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measurable with area $0$.
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\paragraph{Base Case}%
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Consider a set $S$ consisting of a single line segment in a plane.
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By definition of a \lrect{Real.LineSemgnet}{Line Segment}, $S$ is a
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rectangle in which one side has dimension $0$.
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By \larea{Choice-of-Scale}{Choice of Scale}, $S$ is measurable with area its
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width $w$ times its height $h$.
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Therefore $a(S) = wh = 0$.
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\paragraph{Induction Step}%
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Define our induction hypothesis as, "Let $k > 0$ and assume a set consisting
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of $k$ line segments in a plane is measurable with area $0$."
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Consider a set $S_{k+1}$ consisting of $k + 1$ line segments in a plane.
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Pick an arbitrary line segment of $S_{k+1}$.
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Denote the set containing just this line segment as $T$.
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Denote the remaining set of line segments as $S_k$.
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By construction, $S_{k+1} = S_k \cup T$.
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By the induction hypothesis, $S_k$ is measurable with area $0$.
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By the base case, $T$ is measurable with area $0$.
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By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
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measurable, $S_k \cap T$ is measurable, and
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\begin{align}
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a(S_{k+1})
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& = a(S_k \cup T) \nonumber \\
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& = a(S_k) + a(T) - a(S_k \cap T) \nonumber \\
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& = 0 + 0 - a(S_k \cap T). \label{sub:exercise-1c-eq1}
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\end{align}
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\noindent
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There are two cases to consider:
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\subparagraph{Case 1}%
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$S_k \cap T = \emptyset$.
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Then it trivially follows that $a(S_k \cap T) = 0$.
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\subparagraph{Case 2}%
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$S_k \cap T \neq \emptyset$.
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Since $T$ consists of a single point, $S_k \cap T = T$.
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By the base case, $a(S_k \cap T) = a(T) = 0$.
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\vspace{8pt}
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\noindent
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In both cases, \eqref{sub:exercise-1c-eq1} evaluates to $0$, implying
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$a(S_{k+1}) = 0$ as expected.
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\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n > 0$, a set consisting of
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$n$ line segments in a plane is measurable with area $0$.
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\end{proof}
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\section{Exercise 2}%
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\label{sec:exercise-2}
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Every right triangular region is measurable because it can be obtained as the
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intersection of two rectangles. Prove that every triangular region is measurable
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and that its area is one half the product of its base and altitude.
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\begin{proof}
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TODO
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\end{proof}
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\end{document}
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@ -2,8 +2,11 @@
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\input{preamble}
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\newcommand{\link}[1]{\lean{../..}{Exercises/Enderton/Chapter0}
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{Exercises.Enderton.Chapter0.#1}}
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\newcommand{\link}[1]{\lean{../..}
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{Exercises/Enderton/Chapter0}
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{Exercises.Enderton.Chapter0.#1}
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{Chapter0.#1}
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}
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\begin{document}
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@ -15,7 +18,7 @@ Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
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\begin{proof}
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\link{lemma_0a}
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\link{lemma\_0a}
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\end{proof}
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@ -1,5 +1,4 @@
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\usepackage{amsfonts, amsthm}
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\usepackage[T1]{fontenc}
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\usepackage{hyperref}
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\newtheorem{theorem}{Theorem}
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@ -11,11 +10,6 @@
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\hypersetup{colorlinks=true, urlcolor=blue}
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% https://tex.stackexchange.com/a/232188
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\newcommand{\startunderscoreletter}{\catcode`_ 12\relax}
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\newcommand{\stopunderscoreletter}{\catcode`_ 8\relax}
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% The first argument refers to a relative path upward from a current file to
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% the root of the workspace (i.e. where this `preamble.tex` file is located).
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\newcommand{\lean}[3]{\href{#1/#2.html\##3}{
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\startunderscoreletter #3 \stopunderscoreletter}}
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\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}
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