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Add concept of "verified" to statements/theorems.

finite-set-exercises
Joshua Potter 2023-05-10 10:45:42 -06:00
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Bookshelf/Apostol/Chapter_1_07.tex
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@ -1,9 +1,9 @@
\documentclass{article} \documentclass{article}
\usepackage{amsmath}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{mathrsfs}
\input{../../preamble} \input{../../preamble}
\graphicspath{{./images/}} \graphicspath{{./images/}}
\newcommand{\larea}[2]{\lean{../..}{Common/Real/Geometry/Area}{#1}{#2}} \newcommand{\larea}[2]{\lean{../..}{Common/Real/Geometry/Area}{#1}{#2}}
@ -11,16 +11,18 @@
\begin{document} \begin{document}
\header{Exercises 1.7}{Tom M. Apostol}
The properties of area in this set of exercises are to be deduced from the The properties of area in this set of exercises are to be deduced from the
axioms for area stated in the foregoing section. axioms for area stated in the foregoing section.
\section{Exercise 1}% \section*{Exercise 1}%
\label{sec:exercise-1} \hyperlabel{sec:exercise-1}%
Prove that each of the following sets is measurable and has zero area: Prove that each of the following sets is measurable and has zero area:
\subsection{Exercise 1a}% \subsection*{\proceeding{Exercise 1a}}%
\label{sub:exercise-1a} \hyperlabel{sub:exercise-1a}%
A set consisting of a single point. A set consisting of a single point.
@ -36,8 +38,8 @@ A set consisting of a single point.
\end{proof} \end{proof}
\subsection{Exercise 1b}% \subsection*{\proceeding{Exercise 1b}}%
\label{sub:exercise-1b} \hyperlabel{sub:exercise-1b}%
A set consisting of a finite number of points in a plane. A set consisting of a finite number of points in a plane.
@ -50,7 +52,7 @@ A set consisting of a finite number of points in a plane.
\paragraph{Base Case}% \paragraph{Base Case}%
Consider a set $S$ consisting of a single point in a plane. Consider a set $S$ consisting of a single point in a plane.
By \eqref{sub:exercise-1a}, $S$ is measurable with area $0$. By \nameref{sub:exercise-1a}, $S$ is measurable with area $0$.
Thus $P(1)$ holds. Thus $P(1)$ holds.
\paragraph{Induction Step}% \paragraph{Induction Step}%
@ -62,7 +64,7 @@ A set consisting of a finite number of points in a plane.
Denote the remaining set of points as $S_k$. Denote the remaining set of points as $S_k$.
By construction, $S_{k+1} = S_k \cup T$. By construction, $S_{k+1} = S_k \cup T$.
By the induction hypothesis, $S_k$ is measurable with area $0$. By the induction hypothesis, $S_k$ is measurable with area $0$.
By \eqref{sub:exercise-1a}, $T$ is measurable with area $0$. By \nameref{sub:exercise-1a}, $T$ is measurable with area $0$.
By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is By the \larea{Additive-Property}{Additive Property}, $S_k \cup T$ is
measurable, $S_k \cap T$ is measurable, and measurable, $S_k \cap T$ is measurable, and
\begin{align} \begin{align}
@ -82,7 +84,7 @@ A set consisting of a finite number of points in a plane.
$S_k \cap T \neq \emptyset$. $S_k \cap T \neq \emptyset$.
Since $T$ consists of a single point, $S_k \cap T = T$. Since $T$ consists of a single point, $S_k \cap T = T$.
By \eqref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$. By \nameref{sub:exercise-1a}, $a(S_k \cap T) = a(T) = 0$.
\vspace{8pt} \vspace{8pt}
\noindent \noindent
@ -95,8 +97,8 @@ A set consisting of a finite number of points in a plane.
\end{proof} \end{proof}
\subsection{Exercise 1c}% \subsection*{\proceeding{Exercise 1c}}%
\label{sub:exercise-1c} \hyperlabel{sub:exercise-1c}%
The union of a finite collection of line segments in a plane. The union of a finite collection of line segments in a plane.
@ -158,8 +160,8 @@ The union of a finite collection of line segments in a plane.
\end{proof} \end{proof}
\section{Exercise 2}% \section*{\unverified{Exercise 2}}%
\label{sec:exercise-2} \hyperlabel{sec:exercise-2}%
Every right triangular region is measurable because it can be obtained as the Every right triangular region is measurable because it can be obtained as the
intersection of two rectangles. intersection of two rectangles.
@ -209,8 +211,8 @@ Prove that every triangular region is measurable and that its area is one half
\end{proof} \end{proof}
\section{Exercise 3}% \section*{\unverified{Exercise 3}}%
\label{sec:exercise-3} \hyperlabel{sec:exercise-3}%
Prove that every trapezoid and every parallelogram is measurable and derive the Prove that every trapezoid and every parallelogram is measurable and derive the
usual formulas for their areas. usual formulas for their areas.
@ -232,7 +234,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and
height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$. height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$.
By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable. By \larea{Choice-of-Scale}{Choice of Scale}, $R$ is measurable.
By \eqref{sec:exercise-2}, $T$ is measurable. By \nameref{sec:exercise-2}, $T$ is measurable.
By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$ By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
are both measurable and are both measurable and
\begin{align*} \begin{align*}
@ -241,7 +243,8 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
& = a(R) + a(T) - a(R \cap T) \\ & = a(R) + a(T) - a(R \cap T) \\
& = a(R) + a(T) & \text{by construction} \\ & = a(R) + a(T) & \text{by construction} \\
& = b_1h + a(T) & \text{Choice of Scale} \\ & = b_1h + a(T) & \text{Choice of Scale} \\
& = b_1h + \frac{1}{2}(b_2 - b_1)h & \eqref{sec:exercise-2} \\ & = b_1h + \frac{1}{2}(b_2 - b_1)h
& \text{\nameref{sec:exercise-2}} \\
& = \frac{b_1 + b_2}{2}h. & = \frac{b_1 + b_2}{2}h.
\end{align*} \end{align*}
@ -251,7 +254,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$. Then $S$ is the union of non-overlapping triangle $T$ and right trapezoid $R$.
Let $c$ denote the length of base $T$. Let $c$ denote the length of base $T$.
Then $R$ has longer base edge of length $b_2 - c$. Then $R$ has longer base edge of length $b_2 - c$.
By \eqref{sec:exercise-2}, $T$ is measurable. By \nameref{sec:exercise-2}, $T$ is measurable.
By Case 1, $R$ is measurable. By Case 1, $R$ is measurable.
By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$ By the \larea{Additive-Property}{Additive Property}, $R \cup T$ and $R \cap T$
are both measurable and are both measurable and
@ -259,7 +262,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
a(S) a(S)
& = a(T) + a(R) - a(R \cap T) \\ & = a(T) + a(R) - a(R \cap T) \\
& = a(T) + a(R) & \text{by construction} \\ & = a(T) + a(R) & \text{by construction} \\
& = \frac{1}{2}ch + a(R) & \eqref{sec:exercise-2} \\ & = \frac{1}{2}ch + a(R) & \text{\nameref{sec:exercise-2}} \\
& = \frac{1}{2}ch + \frac{b_1 + b_2 - c}{2}h & \text{Case 1} \\ & = \frac{1}{2}ch + \frac{b_1 + b_2 - c}{2}h & \text{Case 1} \\
& = \frac{b_1 + b_2}{2}h. & = \frac{b_1 + b_2}{2}h.
\end{align*} \end{align*}
@ -291,9 +294,7 @@ 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 $$a(S) = \frac{b_1 + b_2}{2}h.$$
\vspace{4pt} \divider
\hrule
\vspace{10pt}
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 $R$.
@ -317,15 +318,15 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
\end{proof} \end{proof}
\section{Exercise 4}% \section*{Exercise 4}%
\label{sec:exercise-4} \hyperlabel{sec:exercise-4}%
Let $P$ be a polygon whose vertices are lattice points. Let $P$ be a polygon whose vertices are lattice points.
The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
lattice points inside the polygon and $B$ denotes the number on the boundary. lattice points inside the polygon and $B$ denotes the number on the boundary.
\subsection{Exercise 4a}% \subsection*{\unverified{Exercise 4a}}%
\label{sub:exercise-4a} \hyperlabel{sub:exercise-4a}%
Prove that the formula is valid for rectangles with sides parallel to the Prove that the formula is valid for rectangles with sides parallel to the
coordinate axes. coordinate axes.
@ -353,8 +354,8 @@ Prove that the formula is valid for rectangles with sides parallel to the
\end{proof} \end{proof}
\subsection{Exercise 4b}% \subsection*{\unverified{Exercise 4b}}%
\label{sub:exercise-4b} \hyperlabel{sub:exercise-4b}%
Prove that the formula is valid for right triangles and parallelograms. Prove that the formula is valid for right triangles and parallelograms.
@ -396,19 +397,19 @@ Prove that the formula is valid for right triangles and parallelograms.
& \eqref{sub:exercise-4b-eq1} \\ & \eqref{sub:exercise-4b-eq1} \\
& = \frac{1}{2}\left[ I_R + \frac{1}{2}B_R - 1 \right] \\ & = \frac{1}{2}\left[ I_R + \frac{1}{2}B_R - 1 \right] \\
& = \frac{1}{2}\left[ (w - 1)(h - 1) + \frac{1}{2}(2(w + h)) - 1 \right] & = \frac{1}{2}\left[ (w - 1)(h - 1) + \frac{1}{2}(2(w + h)) - 1 \right]
& \eqref{sub:exercise-4a} \\ & \text{\nameref{sub:exercise-4a}} \\
& = \frac{1}{2}\left[ (w - 1)(h - 1) + w + h - 1 \right] \\ & = \frac{1}{2}\left[ (w - 1)(h - 1) + w + h - 1 \right] \\
& = \frac{1}{2}\left[ wh - w - h + 1 + w + h - 1 \right] \\ & = \frac{1}{2}\left[ wh - w - h + 1 + w + h - 1 \right] \\
& = \frac{wh}{2}. & = \frac{wh}{2}.
\end{align*} \end{align*}
We do not prove this formula is valid for parallelograms here. We do not prove this formula is valid for parallelograms here.
Instead, refer to \eqref{sub:exercise-4c} below. Instead, refer to \nameref{sub:exercise-4c} below.
\end{proof} \end{proof}
\subsection{Exercise 4c}% \subsection*{\unverified{Exercise 4c}}%
\label{sub:exercise-4c} \hyperlabel{sub:exercise-4c}%
Use induction on the number of edges to construct a proof for general polygons. Use induction on the number of edges to construct a proof for general polygons.
@ -421,7 +422,7 @@ Use induction on the number of edges to construct a proof for general polygons.
\paragraph{Base Case}% \paragraph{Base Case}%
A $3$-polygon is a triangle. A $3$-polygon is a triangle.
By \eqref{sub:exercise-4b}, the lattice point area formula holds. By \nameref{sub:exercise-4b}, the lattice point area formula holds.
Thus $P(3)$ holds. Thus $P(3)$ holds.
\paragraph{Induction Step}% \paragraph{Induction Step}%
@ -473,8 +474,8 @@ Use induction on the number of edges to construct a proof for general polygons.
\end{proof} \end{proof}
\section{Exercise 5}% \section*{\unverified{Exercise 5}}%
\label{sec:exercise-5} \hyperlabel{sec:exercise-5}%
Prove that a triangle whose vertices are lattice points cannot be equilateral. Prove that a triangle whose vertices are lattice points cannot be equilateral.
@ -487,7 +488,7 @@ ways, using Exercises 2 and 4.]
Let $T$ be an equilateral triangle whose vertices are lattice points. Let $T$ be an equilateral triangle whose vertices are lattice points.
Assume each side of $T$ has length $a$. Assume each side of $T$ has length $a$.
Then $T$ has height $h = (a\sqrt{3}) / 2$. Then $T$ has height $h = (a\sqrt{3}) / 2$.
By \eqref{sec:exercise-2}, By \nameref{sec:exercise-2},
\begin{equation} \begin{equation}
\label{sub:exercise-5-eq1} \label{sub:exercise-5-eq1}
\tag{5.1} \tag{5.1}
@ -495,7 +496,7 @@ ways, using Exercises 2 and 4.]
\end{equation} \end{equation}
Let $I$ and $B$ denote the number of interior and boundary lattice points of Let $I$ and $B$ denote the number of interior and boundary lattice points of
$T$ respectively. $T$ respectively.
By \eqref{sec:exercise-4}, By \nameref{sec:exercise-4},
\begin{equation} \begin{equation}
\label{sub:exercise-5-eq2} \label{sub:exercise-5-eq2}
\tag{5.2} \tag{5.2}
@ -509,8 +510,8 @@ ways, using Exercises 2 and 4.]
\end{proof} \end{proof}
\section{Exercise 6}% \section*{\unverified{Exercise 6}}%
\label{sec:exercise-6} \hyperlabel{sec:exercise-6}%
Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
subsets of $A$. subsets of $A$.

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Bookshelf/Apostol/Chapter_1_11.tex
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@ -1,25 +1,24 @@
\documentclass{article} \documentclass{article}
\usepackage{amsmath}
\usepackage[shortlabels]{enumitem}
\usepackage{soul, xcolor}
\input{../../preamble} \input{../../preamble}
\newcommand{\link}[1]{\lean{../..} \newcommand{\link}[1]{\lean{../..}
{Bookshelf/Apostol/Chapter\_1\_11} {Bookshelf/Apostol/Chapter\_1\_11} % Location
{Apostol.Chapter\_1\_11.#1} {Apostol.Chapter\_1\_11.#1} % Namespace
{Chapter\_1\_11.#1} {Chapter\_1\_11.#1} % Presentation
} }
\begin{document} \begin{document}
\section{Exercise 4}% \header{Exercises 1.11}{Tom M. Apostol}
\label{sec:exercise-4}
\section*{Exercise 4}%
\hyperlabel{sec:exercise-4}%
Prove that the greatest-integer function has the properties indicated: Prove that the greatest-integer function has the properties indicated:
\subsection{Exercise 4a}% \subsection*{\proceeding{Exercise 4a}}%
\label{sub:exercise-4a} \hyperlabel{sub:exercise-4a}%
$\floor{x + n} = \floor{x} + n$ for every integer $n$. $\floor{x + n} = \floor{x} + n$ for every integer $n$.
@ -29,8 +28,8 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
\end{proof} \end{proof}
\subsection{Exercise 4b}% \subsection*{\proceeding{Exercise 4b}}%
\label{sub:exercise-4b} \hyperlabel{sub:exercise-4b}%
$\floor{-x} = $\floor{-x} =
\begin{cases} \begin{cases}
@ -49,8 +48,8 @@ $\floor{-x} =
\end{proof} \end{proof}
\subsection{Exercise 4c}% \subsection*{\proceeding{Exercise 4c}}%
\label{sub:exercise-4c} \hyperlabel{sub:exercise-4c}%
$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$. $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
@ -60,8 +59,8 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
\end{proof} \end{proof}
\subsection{Exercise 4d}% \subsection*{\proceeding{Exercise 4d}}%
\label{sub:exercise-4d} \hyperlabel{sub:exercise-4d}%
$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
@ -71,8 +70,8 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
\end{proof} \end{proof}
\subsection{Exercise 4e}% \subsection*{\proceeding{Exercise 4e}}%
\label{sub:exercise-4e} \hyperlabel{sub:exercise-4e}%
$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
@ -82,24 +81,20 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
\end{proof} \end{proof}
\section{Exercise 5}% \section*{\proceeding{Exercise 5}}%
\label{sec:exercise-5} \hyperlabel{sec:exercise-5}%
The formulas in Exercises 4(d) and 4(e) suggest a generalization for The formulas in Exercises 4(d) and 4(e) suggest a generalization for
$\floor{nx}$. $\floor{nx}$.
State and prove such a generalization. State and prove such a generalization.
\vspace{6pt} \note{The stated generalization is known as "Hermite's Identity."}
\noindent
\hl{Note}: The stated generalization is known as "Hermite's Identity."
\begin{proof} \begin{proof}
\link{exercise\_5} \link{exercise\_5}
\vspace{10pt} \divider
\hrule
\vspace{10pt}
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:
@ -132,7 +127,7 @@ State and prove such a generalization.
& = \floor{n(\floor{x} + r)} \nonumber \\ & = \floor{n(\floor{x} + r)} \nonumber \\
& = \floor{n\floor{x} + nr} \nonumber \\ & = \floor{n\floor{x} + nr} \nonumber \\
& = \floor{n\floor{x}} + \floor{nr}. \nonumber & = \floor{n\floor{x}} + \floor{nr}. \nonumber
& \eqref{sub:exercise-4a} \\ & \text{\nameref{sub:exercise-4a}} \\
& = \floor{n\floor{x}} + j \nonumber \\ & = \floor{n\floor{x}} + j \nonumber \\
& = n\floor{x} + j. \label{sec:exercise-5-eq3} & = n\floor{x} + j. \label{sec:exercise-5-eq3}
\end{align} \end{align}
@ -175,8 +170,8 @@ State and prove such a generalization.
\end{proof} \end{proof}
\section{Exercise 6}% \section*{\unverified{Exercise 6}}%
\label{sec:exercise-6} \hyperlabel{sec:exercise-6}%
Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
integers. integers.
@ -199,15 +194,15 @@ Prove that the number of lattice points in $S$ is equal to the sum
\end{proof} \end{proof}
\section{Exercise 7}% \section*{Exercise 7}%
\label{sec:exercise-7} \hyperlabel{sec:exercise-7}%
If $a$ and $b$ are positive integers with no common factor, we have the formula If $a$ and $b$ are positive integers with no common factor, we have the formula
$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$ $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
When $b = 1$, the sum on the left is understood to be $0$. When $b = 1$, the sum on the left is understood to be $0$.
\subsection{Exercise 7a}% \subsection*{\unverified{Exercise 7a}}%
\label{sub:exercise-7a} \hyperlabel{sub:exercise-7a}%
Derive this result by a geometric argument, counting lattice points in a right Derive this result by a geometric argument, counting lattice points in a right
triangle. triangle.
@ -218,8 +213,8 @@ Derive this result by a geometric argument, counting lattice points in a right
\end{proof} \end{proof}
\subsection{Exercise 7b}% \subsection*{\proceeding{Exercise 7b}}%
\label{sub:exercise-7b} \hyperlabel{sub:exercise-7b}%
Derive the result analytically as follows: Derive the result analytically as follows:
By changing the index of summation, note that By changing the index of summation, note that
@ -232,8 +227,8 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
\end{proof} \end{proof}
\section{Exercise 8}% \section*{\unverified{Exercise 8}}%
\label{sec:exercise-8} \hyperlabel{sec:exercise-8}%
Let $S$ be a set of points on the real line. Let $S$ be a set of points on the real line.
The \textit{characteristic function} of $S$ is, by definition, the function The \textit{characteristic function} of $S$ is, by definition, the function

35
Bookshelf/Apostol/Chapter_I_03.tex
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@ -1,5 +1,4 @@
\documentclass{article} \documentclass{article}
\usepackage[shortlabels]{enumitem}
\input{../../preamble} \input{../../preamble}
@ -11,8 +10,10 @@
\begin{document} \begin{document}
\section*{Theorem I.27}% \header{A Set of Axioms for the Real-Number System}{Tom M. Apostol}
\label{sec:theorem-i.27}
\section*{\proceeding{Theorem I.27}}%
\hyperlabel{sec:theorem-i.27}%
Every nonempty set $S$ that is bounded below has a greatest lower bound; that Every nonempty set $S$ that is bounded below has a greatest lower bound; that
is, there is a real number $L$ such that $L = \inf{S}$. is, there is a real number $L$ such that $L = \inf{S}$.
@ -23,8 +24,8 @@ is, there is a real number $L$ such that $L = \inf{S}$.
\end{proof} \end{proof}
\section*{Theorem I.29}% \section*{\proceeding{Theorem I.29}}%
\label{sec:theorem-i.29} \hyperlabel{sec:theorem-i.29}
For every real $x$ there exists a positive integer $n$ such that $n > x$. For every real $x$ there exists a positive integer $n$ such that $n > x$.
@ -34,20 +35,22 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
\end{proof} \end{proof}
\section*{Theorem I.30 (Archimedean Property of the Reals)}% \section*{\proceeding{Theorem I.30}}%
\label{sec:theorem-i.30} \hyperlabel{sec:theorem-i.30}%
If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
integer $n$ such that $nx > y$. integer $n$ such that $nx > y$.
\note{This is known as the "Archimedean Property of the Reals."}
\begin{proof} \begin{proof}
\link{exists\_pnat\_mul\_self\_geq\_of\_pos} \link{exists\_pnat\_mul\_self\_geq\_of\_pos}
\end{proof} \end{proof}
\section*{Theorem I.31}% \section*{\proceeding{Theorem I.31}}%
\label{sec:theorem-i.31} \hyperlabel{sec:theorem-i.31}%
If three real numbers $a$, $x$, and $y$ satisfy the inequalities If three real numbers $a$, $x$, and $y$ satisfy the inequalities
$$a \leq x \leq a + \frac{y}{n}$$ $$a \leq x \leq a + \frac{y}{n}$$
@ -59,8 +62,8 @@ for every integer $n \geq 1$, then $x = a$.
\end{proof} \end{proof}
\section*{Theorem I.32}% \section*{\proceeding{Theorem I.32}}%
\label{sec:theorem-i.32} \hyperlabel{sec:theorem-i.32}%
Let $h$ be a given positive number and let $S$ be a set of real numbers. Let $h$ be a given positive number and let $S$ be a set of real numbers.
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
@ -81,8 +84,8 @@ Let $h$ be a given positive number and let $S$ be a set of real numbers.
\end{proof} \end{proof}
\section*{Theorem I.33 (Additive Property)}% \section*{\proceeding{Theorem I.33}}%
\label{sec:theorem-i.33} \hyperlabel{sec:theorem-i.33}%
Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
$$C = \{a + b : a \in A, b \in B\}.$$ $$C = \{a + b : a \in A, b \in B\}.$$
@ -94,6 +97,8 @@ $$C = \{a + b : a \in A, b \in B\}.$$
$$\inf{C} = \inf{A} + \inf{B}.$$ $$\inf{C} = \inf{A} + \inf{B}.$$
\end{enumerate} \end{enumerate}
\note{This is known as the "Additive Property."}
\begin{proof} \begin{proof}
\ % Force space prior to *Proof.* \ % Force space prior to *Proof.*
@ -105,8 +110,8 @@ $$C = \{a + b : a \in A, b \in B\}.$$
\end{proof} \end{proof}
\section*{Theorem I.34}% \section*{\proceeding{Theorem I.34}}%
\label{sec:theorem-i.34} \hyperlabel{sec:theorem-i.34}%
Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that
$$s \leq t$$ $$s \leq t$$

15
Bookshelf/Enderton/Chapter0.tex
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@ -3,17 +3,20 @@
\input{../../preamble} \input{../../preamble}
\newcommand{\link}[1]{\lean{../..} \newcommand{\link}[1]{\lean{../..}
{Bookshelf/Enderton/Chapter0} {Bookshelf/Enderton/Chapter0} % Location
{Enderton.Chapter0.#1} {Enderton.Chapter0.#1} % Namespace
{Chapter0.#1} {Chapter0.#1} % Presentation
} }
\begin{document} \begin{document}
\section*{Lemma 0A}% \header{Useful Facts About Sets}{Herbert B. Enderton}
\label{sec:lemma-0a}
Assume that $\langle x_1, \ldots, x_m \rangle = \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$. \section{\proceeding{Lemma 0A}}%
\hyperlabel{sec:lemma-0a}%
Assume that $\langle x_1, \ldots, x_m \rangle =
\langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.
Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$. Then $x_1 = \langle y_1, \ldots, y_{k+1} \rangle$.
\begin{proof} \begin{proof}

10
Common/Real/Sequence/Arithmetic.tex
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@ -3,15 +3,15 @@
\input{../../../preamble} \input{../../../preamble}
\newcommand{\link}[1]{\lean{../../..} \newcommand{\link}[1]{\lean{../../..}
{Common/Real/Sequence/Arithmetic} {Common/Real/Sequence/Arithmetic} % Location
{Real.Arithmetic.#1} {Real.Arithmetic.#1} % Namespace
{Real.Arithmetic.#1} {Real.Arithmetic.#1} % Presentation
} }
\begin{document} \begin{document}
\section*{Sum of Arithmetic Series}% \section{\proceeding{Sum of Arithmetic Series}}%
\label{sec:sum-arithmetic-series} \hyperlabel{sec:sum-arithmetic-series}%
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 $d$.
Then for some $n \in \mathbb{N}$, Then for some $n \in \mathbb{N}$,

10
Common/Real/Sequence/Geometric.tex
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@ -3,15 +3,15 @@
\input{../../../preamble} \input{../../../preamble}
\newcommand{\link}[1]{\lean{../../..} \newcommand{\link}[1]{\lean{../../..}
{Common/Real/Sequence/Geometric} {Common/Real/Sequence/Geometric} % Location
{Real.Geometric.#1} {Real.Geometric.#1} % Namespace
{Real.Geometric.#1} {Real.Geometric.#1} % Presentation
} }
\begin{document} \begin{document}
\section*{Sum of Geometric Series}% \section{\proceeding{Sum of Geometric Series}}%
\label{sec:sum-geometric-series} \hyperlabel{sec:sum-geometric-series}%
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 $r \neq 1$.
Then for some $n \in \mathbb{N}$, Then for some $n \in \mathbb{N}$,

68
preamble.tex
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@ -1,15 +1,66 @@
\usepackage{amsfonts, amsthm} \usepackage{amsfonts, amsmath, amsthm}
\usepackage[shortlabels]{enumitem}
\usepackage{environ}
\usepackage{fancybox}
\usepackage{fontawesome5}
\usepackage{hyperref} \usepackage{hyperref}
\usepackage{mathrsfs}
\usepackage{soul, xcolor}
\newtheorem{theorem}{Theorem} % ========================================
\newtheorem{xtheoreminner}{Theorem} % Linking
\newenvironment{xtheorem}[1]{% % ========================================
\renewcommand\thextheoreminner{#1}%
\xtheoreminner
}{\endxtheoreminner}
\hypersetup{colorlinks=true, urlcolor=blue} \hypersetup{colorlinks=true, urlcolor=blue}
% The first argument refers to a relative path upward from a current file to
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% #1 - Path to root
% #2 - Location
% #3 - Namespace
% #4 - Presentation
\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}
\newcommand{\hyperlabel}[1]{%
\label{#1}%
\hypertarget{#1}{}}
% ========================================
% Environments
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\author{#2}
\date{}
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\begin{center}
\doublebox{
\begin{minipage}{0.95\textwidth}
\vspace{2pt}
\hl{Note}: #1
\vspace{2pt}
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% Status of a proof. A statement/theorem is verified if both a LaTeX proof
% and a corresponding Lean proof has been written. If a Lean proof is in
% progress, it's in a "proceeding" state. Otherwise it is unverified.
\DeclareRobustCommand{\verified}[1]{%
\texorpdfstring{\color{teal}#1\ \faCheckCircle}{#1}}
\DeclareRobustCommand{\proceeding}[1]{%
\texorpdfstring{\color{magenta}#1\ \faSpinner}{#1}}
\DeclareRobustCommand{\unverified}[1]{%
\texorpdfstring{\color{red}#1\ \faExclamationCircle}{#1}}
% ========================================
% Math
% ========================================
\newcommand{\abs}[1]{\left|#1\right|} \newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
@ -17,6 +68,3 @@
\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)}
% The first argument refers to a relative path upward from a current file to
% the root of the workspace (i.e. where this `preamble.tex` file is located).
\newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}