Remove no longer needed `hyperlabel` command.

Bookshelf/Apostol/Chapter_1_07.tex

@@ -18,12 +18,12 @@ The properties of area in this set of exercises are to be deduced from the
   axioms for area stated in the foregoing section.
 
 \section*{Exercise 1}%
-\hyperlabel{sec:exercise-1}%
+\label{sec:exercise-1}
 
 Prove that each of the following sets is measurable and has zero area:
 
 \subsection*{\proceeding{Exercise 1a}}%
-\hyperlabel{sub:exercise-1a}%
+\label{sub:exercise-1a}
 
 A set consisting of a single point.
 
@@ -39,7 +39,7 @@ A set consisting of a single point.
 \end{proof}
 
 \subsection*{\proceeding{Exercise 1b}}%
-\hyperlabel{sub:exercise-1b}%
+\label{sub:exercise-1b}
 
 A set consisting of a finite number of points in a plane.
 
@@ -98,7 +98,7 @@ A set consisting of a finite number of points in a plane.
 \end{proof}
 
 \subsection*{\proceeding{Exercise 1c}}%
-\hyperlabel{sub:exercise-1c}%
+\label{sub:exercise-1c}
 
 The union of a finite collection of line segments in a plane.
 
@@ -161,7 +161,7 @@ The union of a finite collection of line segments in a plane.
 \end{proof}
 
 \section*{\unverified{Exercise 2}}%
-\hyperlabel{sec:exercise-2}%
+\label{sec:exercise-2}
 
 Every right triangular region is measurable because it can be obtained as the
   intersection of two rectangles.
@@ -212,7 +212,7 @@ Prove that every triangular region is measurable and that its area is one half
 \end{proof}
 
 \section*{\unverified{Exercise 3}}%
-\hyperlabel{sec:exercise-3}%
+\label{sec:exercise-3}
 
 Prove that every trapezoid and every parallelogram is measurable and derive the
   usual formulas for their areas.
@@ -319,14 +319,14 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
 \end{proof}
 
 \section*{Exercise 4}%
-\hyperlabel{sec:exercise-4}%
+\label{sec:exercise-4}
 
 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
   lattice points inside the polygon and $B$ denotes the number on the boundary.
 
 \subsection*{\unverified{Exercise 4a}}%
-\hyperlabel{sub:exercise-4a}%
+\label{sub:exercise-4a}
 
 Prove that the formula is valid for rectangles with sides parallel to the
   coordinate axes.
@@ -354,7 +354,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
 \end{proof}
 
 \subsection*{\unverified{Exercise 4b}}%
-\hyperlabel{sub:exercise-4b}%
+\label{sub:exercise-4b}
 
 Prove that the formula is valid for right triangles and parallelograms.
 
@@ -408,7 +408,7 @@ Prove that the formula is valid for right triangles and parallelograms.
 \end{proof}
 
 \subsection*{\unverified{Exercise 4c}}%
-\hyperlabel{sub:exercise-4c}%
+\label{sub:exercise-4c}
 
 Use induction on the number of edges to construct a proof for general polygons.
 
@@ -474,7 +474,7 @@ Use induction on the number of edges to construct a proof for general polygons.
 \end{proof}
 
 \section*{\unverified{Exercise 5}}%
-\hyperlabel{sec:exercise-5}%
+\label{sec:exercise-5}
 
 Prove that a triangle whose vertices are lattice points cannot be equilateral.
 
@@ -510,7 +510,7 @@ ways, using Exercises 2 and 4.]
 \end{proof}
 
 \section*{\unverified{Exercise 6}}%
-\hyperlabel{sec:exercise-6}%
+\label{sec:exercise-6}
 
 Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
   subsets of $A$.

Bookshelf/Apostol/Chapter_1_11.tex

@@ -11,12 +11,12 @@
 \header{Exercises 1.11}{Tom M. Apostol}
 
 \section*{Exercise 4}%
-\hyperlabel{sec:exercise-4}%
+\label{sec:exercise-4}
 
 Prove that the greatest-integer function has the properties indicated:
 
 \subsection*{\proceeding{Exercise 4a}}%
-\hyperlabel{sub:exercise-4a}%
+\label{sub:exercise-4a}
 
 $\floor{x + n} = \floor{x} + n$ for every integer $n$.
 
@@ -27,7 +27,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
 \end{proof}
 
 \subsection*{\proceeding{Exercise 4b}}%
-\hyperlabel{sub:exercise-4b}%
+\label{sub:exercise-4b}
 
 $\floor{-x} =
   \begin{cases}
@@ -47,7 +47,7 @@ $\floor{-x} =
 \end{proof}
 
 \subsection*{\proceeding{Exercise 4c}}%
-\hyperlabel{sub:exercise-4c}%
+\label{sub:exercise-4c}
 
 $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
 
@@ -58,7 +58,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
 \end{proof}
 
 \subsection*{\proceeding{Exercise 4d}}%
-\hyperlabel{sub:exercise-4d}%
+\label{sub:exercise-4d}
 
 $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
 
@@ -69,7 +69,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
 \end{proof}
 
 \subsection*{\proceeding{Exercise 4e}}%
-\hyperlabel{sub:exercise-4e}%
+\label{sub:exercise-4e}
 
 $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
 
@@ -80,7 +80,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
 \end{proof}
 
 \section*{\proceeding{Exercise 5}}%
-\hyperlabel{sec:exercise-5}%
+\label{sec:exercise-5}
 
 The formulas in Exercises 4(d) and 4(e) suggest a generalization for
   $\floor{nx}$.
@@ -169,7 +169,7 @@ State and prove such a generalization.
 \end{proof}
 
 \section*{\unverified{Exercise 6}}%
-\hyperlabel{sec:exercise-6}%
+\label{sec:exercise-6}
 
 Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
   integers.
@@ -193,14 +193,14 @@ Prove that the number of lattice points in $S$ is equal to the sum
 \end{proof}
 
 \section*{Exercise 7}%
-\hyperlabel{sec:exercise-7}%
+\label{sec:exercise-7}
 
 If $a$ and $b$ are positive integers with no common factor, we have the formula
   $$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
 When $b = 1$, the sum on the left is understood to be $0$.
 
 \subsection*{\unverified{Exercise 7a}}%
-\hyperlabel{sub:exercise-7a}%
+\label{sub:exercise-7a}
 
 Derive this result by a geometric argument, counting lattice points in a right
   triangle.
@@ -212,7 +212,7 @@ Derive this result by a geometric argument, counting lattice points in a right
 \end{proof}
 
 \subsection*{\proceeding{Exercise 7b}}%
-\hyperlabel{sub:exercise-7b}%
+\label{sub:exercise-7b}
 
 Derive the result analytically as follows:
 By changing the index of summation, note that
@@ -226,7 +226,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
 \end{proof}
 
 \section*{\unverified{Exercise 8}}%
-\hyperlabel{sec:exercise-8}%
+\label{sec:exercise-8}
 
 Let $S$ be a set of points on the real line.
 The \textit{characteristic function} of $S$ is, by definition, the function

Bookshelf/Apostol/Chapter_I_03.tex

@@ -11,7 +11,7 @@
 \header{A Set of Axioms for the Real-Number System}{Tom M. Apostol}
 
 \section*{\verified{Lemma 1}}%
-\hyperlabel{sec:lemma-1}%
+\label{sec:lemma-1}
 
 Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
 
@@ -31,7 +31,7 @@ Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
 \end{proof}
 
 \section*{\verified{Theorem I.27}}%
-\hyperlabel{sec:theorem-i.27}%
+\label{sec:theorem-i.27}
 
 Every nonempty set $S$ that is bounded below has a greatest lower bound; that
   is, there is a real number $L$ such that $L = \inf{S}$.
@@ -51,7 +51,7 @@ Every nonempty set $S$ that is bounded below has a greatest lower bound; that
 \end{proof}
 
 \section*{\verified{Theorem I.29}}%
-\hyperlabel{sec:theorem-i.29}
+\label{sec:theorem-i.29}
 
 For every real $x$ there exists a positive integer $n$ such that $n > x$.
 
@@ -71,7 +71,7 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
 \end{proof}
 
 \section*{\verified{Theorem I.30}}%
-\hyperlabel{sec:theorem-i.30}%
+\label{sec:theorem-i.30}
 
 If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
   integer $n$ such that $nx > y$.
@@ -92,7 +92,7 @@ If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
 \end{proof}
 
 \section*{\verified{Theorem I.31}}%
-\hyperlabel{sec:theorem-i.31}%
+\label{sec:theorem-i.31}
 
 If three real numbers $a$, $x$, and $y$ satisfy the inequalities
   $$a \leq x \leq a + \frac{y}{n}$$ for every integer $n \geq 1$, then $x = a$.
@@ -135,7 +135,7 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
 \end{proof}
 
 \section*{\verified{Lemma 2}}%
-\hyperlabel{sec:lemma-2}%
+\label{sec:lemma-2}
 
 If three real numbers $a$, $x$, and $y$ satisfy the inequalities
   $$a - y / n \leq x \leq a$$ for every integer $n \geq 1$, then $x = a$.
@@ -178,12 +178,12 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
 \end{proof}
 
 \section*{Theorem I.32}%
-\hyperlabel{sec:theorem-i.32}%
+\label{sec:theorem-i.32}
 
 Let $h$ be a given positive number and let $S$ be a set of real numbers.
 
 \subsection*{\verified{Theorem I.32a}}%
-\hyperlabel{sub:theorem-i.32a}%
+\label{sub:theorem-i.32a}
 
 If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
 
@@ -205,7 +205,7 @@ If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
 \end{proof}
 
 \subsection*{\verified{Theorem I.32b}}%
-\hyperlabel{sub:theorem-i.32b}%
+\label{sub:theorem-i.32b}
 
 If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
 
@@ -227,7 +227,7 @@ If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
 \end{proof}
 
 \section*{Theorem I.33}%
-\hyperlabel{sec:theorem-i.33}%
+\label{sec:theorem-i.33}
 
 Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
   $$C = \{a + b : a \in A, b \in B\}.$$
@@ -235,7 +235,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
 \note{This is known as the "Additive Property."}
 
 \subsection*{\verified{Theorem I.33a}}%
-\hyperlabel{sub:theorem-i.33a}%
+\label{sub:theorem-i.33a}
 
 If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
   $$\sup{C} = \sup{A} + \sup{B}.$$
@@ -250,7 +250,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
     $\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
 
   \paragraph{(i)}%
-  \hyperlabel{par:theorem-i.33a-i}%
+  \label{par:theorem-i.33a-i}
 
     Let $x \in C$.
     By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
@@ -303,7 +303,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
 \end{proof}
 
 \subsection*{\verified{Theorem I.33b}}%
-\hyperlabel{sub:theorem-i.33b}%
+\label{sub:theorem-i.33b}
 
 If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
   $$\inf{C} = \inf{A} + \inf{B}.$$
@@ -318,7 +318,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
     $\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
 
   \paragraph{(i)}%
-  \hyperlabel{par:theorem-i.33b-i}%
+  \label{par:theorem-i.33b-i}
 
     Let $x \in C$.
     By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
@@ -371,7 +371,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
 \end{proof}
 
 \section*{\verified{Theorem I.34}}%
-\hyperlabel{sec:theorem-i.34}%
+\label{sec:theorem-i.34}
 
 Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that $$s \leq t$$
   for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$

Bookshelf/Enderton/Chapter_0.tex

@@ -11,7 +11,7 @@
 \header{Useful Facts About Sets}{Herbert B. Enderton}
 
 \section*{\proceeding{Lemma 0A}}%
-\hyperlabel{sec:lemma-0a}%
+\label{sec:lemma-0a}
 
 Assume that $\langle x_1, \ldots, x_m \rangle =
   \langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.

Common/Real/Geometry/Area.tex

@@ -13,7 +13,7 @@ We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and
   properties:
 
 \section*{\defined{Nonnegative Property}}%
-\hyperlabel{sec:nonnegative-property}%
+\label{sec:nonnegative-property}
 
 For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
 
@@ -24,7 +24,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
 \end{axiom}
 
 \section*{\defined{Additive Property}}%
-\hyperlabel{sec:additive-property}%
+\label{sec:additive-property}
 
 If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
   $\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
@@ -36,7 +36,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
 \end{axiom}
 
 \section*{\defined{Difference Property}}%
-\hyperlabel{sec:difference-property}%
+\label{sec:difference-property}
 
 If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
   $\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
@@ -48,7 +48,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
 \end{axiom}
 
 \section*{\defined{Invariance Under Congruence}}%
-\hyperlabel{sec:invariance-under-congruence}%
+\label{sec:invariance-under-congruence}
 
 If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
   also in $\mathscr{M}$ and we have $a(S) = a(T)$.
@@ -72,7 +72,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
 \end{axiom}
 
 \section*{\proceeding{Exhaustion Property}}%
-\hyperlabel{sec:exhaustion-property}%
+\label{sec:exhaustion-property}
 
 Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
   that

Common/Real/Sequence/Arithmetic.tex

@@ -9,7 +9,7 @@
 \begin{document}
 
 \section{\proceeding{Sum of Arithmetic Series}}%
-\hyperlabel{sec:sum-arithmetic-series}%
+\label{sec:sum-arithmetic-series}
 
 Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
 Then for some $n \in \mathbb{N}$,

Common/Real/Sequence/Geometric.tex

@@ -9,7 +9,7 @@
 \begin{document}
 
 \section{\proceeding{Sum of Geometric Series}}%
-\hyperlabel{sec:sum-geometric-series}%
+\label{sec:sum-geometric-series}
 
 Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
 Then for some $n \in \mathbb{N}$,

preamble.tex

@@ -16,9 +16,6 @@
 
 \hypersetup{colorlinks=true, urlcolor=blue}
 \newcommand{\leanref}[2]{\color{blue}$\pmb{\exists}\;{-}\;$\href{#1}{#2}}
-\newcommand{\hyperlabel}[1]{%
-  \label{#1}%
-  \hypertarget{#1}{}}
 
 % ========================================
 % Environments