Remove no longer needed `hyperlabel` command.
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@ -18,12 +18,12 @@ 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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\hyperlabel{sec: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*{\proceeding{Exercise 1a}}%
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\hyperlabel{sub: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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@ -39,7 +39,7 @@ A set consisting of a single point.
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\end{proof}
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\subsection*{\proceeding{Exercise 1b}}%
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\hyperlabel{sub: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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@ -98,7 +98,7 @@ A set consisting of a finite number of points in a plane.
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\end{proof}
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\subsection*{\proceeding{Exercise 1c}}%
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\hyperlabel{sub: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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@ -161,7 +161,7 @@ The union of a finite collection of line segments in a plane.
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\end{proof}
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\section*{\unverified{Exercise 2}}%
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\hyperlabel{sec: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.
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@ -212,7 +212,7 @@ Prove that every triangular region is measurable and that its area is one half
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\end{proof}
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\section*{\unverified{Exercise 3}}%
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\hyperlabel{sec:exercise-3}%
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\label{sec:exercise-3}
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Prove that every trapezoid and every parallelogram is measurable and derive the
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usual formulas for their areas.
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@ -319,14 +319,14 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
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\end{proof}
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\section*{Exercise 4}%
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\hyperlabel{sec:exercise-4}%
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\label{sec:exercise-4}
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Let $P$ be a polygon whose vertices are lattice points.
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The area of $P$ is $I + \frac{1}{2}B - 1$, where $I$ denotes the number of
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lattice points inside the polygon and $B$ denotes the number on the boundary.
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\subsection*{\unverified{Exercise 4a}}%
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\hyperlabel{sub:exercise-4a}%
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\label{sub:exercise-4a}
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Prove that the formula is valid for rectangles with sides parallel to the
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coordinate axes.
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@ -354,7 +354,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
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\end{proof}
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\subsection*{\unverified{Exercise 4b}}%
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\hyperlabel{sub:exercise-4b}%
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\label{sub:exercise-4b}
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Prove that the formula is valid for right triangles and parallelograms.
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@ -408,7 +408,7 @@ Prove that the formula is valid for right triangles and parallelograms.
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\end{proof}
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\subsection*{\unverified{Exercise 4c}}%
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\hyperlabel{sub:exercise-4c}%
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\label{sub:exercise-4c}
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Use induction on the number of edges to construct a proof for general polygons.
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@ -474,7 +474,7 @@ Use induction on the number of edges to construct a proof for general polygons.
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\end{proof}
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\section*{\unverified{Exercise 5}}%
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\hyperlabel{sec:exercise-5}%
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\label{sec:exercise-5}
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Prove that a triangle whose vertices are lattice points cannot be equilateral.
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@ -510,7 +510,7 @@ ways, using Exercises 2 and 4.]
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\end{proof}
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\section*{\unverified{Exercise 6}}%
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\hyperlabel{sec:exercise-6}%
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\label{sec:exercise-6}
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Let $A = \{1, 2, 3, 4, 5\}$, and let $\mathscr{M}$ denote the class of all
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subsets of $A$.
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@ -11,12 +11,12 @@
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\header{Exercises 1.11}{Tom M. Apostol}
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\section*{Exercise 4}%
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\hyperlabel{sec:exercise-4}%
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\label{sec:exercise-4}
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Prove that the greatest-integer function has the properties indicated:
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\subsection*{\proceeding{Exercise 4a}}%
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\hyperlabel{sub:exercise-4a}%
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\label{sub:exercise-4a}
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$\floor{x + n} = \floor{x} + n$ for every integer $n$.
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@ -27,7 +27,7 @@ $\floor{x + n} = \floor{x} + n$ for every integer $n$.
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\end{proof}
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\subsection*{\proceeding{Exercise 4b}}%
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\hyperlabel{sub:exercise-4b}%
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\label{sub:exercise-4b}
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$\floor{-x} =
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\begin{cases}
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@ -47,7 +47,7 @@ $\floor{-x} =
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\end{proof}
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\subsection*{\proceeding{Exercise 4c}}%
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\hyperlabel{sub:exercise-4c}%
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\label{sub:exercise-4c}
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$\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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@ -58,7 +58,7 @@ $\floor{x + y} = \floor{x} + \floor{y}$ or $\floor{x} + \floor{y} + 1$.
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\end{proof}
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\subsection*{\proceeding{Exercise 4d}}%
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\hyperlabel{sub:exercise-4d}%
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\label{sub:exercise-4d}
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$\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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@ -69,7 +69,7 @@ $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}.$
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\end{proof}
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\subsection*{\proceeding{Exercise 4e}}%
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\hyperlabel{sub:exercise-4e}%
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\label{sub:exercise-4e}
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$\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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@ -80,7 +80,7 @@ $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{2}{3}}.$
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\end{proof}
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\section*{\proceeding{Exercise 5}}%
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\hyperlabel{sec:exercise-5}%
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\label{sec:exercise-5}
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The formulas in Exercises 4(d) and 4(e) suggest a generalization for
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$\floor{nx}$.
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@ -169,7 +169,7 @@ State and prove such a generalization.
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\end{proof}
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\section*{\unverified{Exercise 6}}%
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\hyperlabel{sec:exercise-6}%
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\label{sec:exercise-6}
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Recall that a lattice point $(x, y)$ in the plane is one whose coordinates are
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integers.
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@ -193,14 +193,14 @@ Prove that the number of lattice points in $S$ is equal to the sum
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\end{proof}
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\section*{Exercise 7}%
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\hyperlabel{sec:exercise-7}%
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\label{sec:exercise-7}
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If $a$ and $b$ are positive integers with no common factor, we have the formula
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$$\sum_{n=1}^{b-1} \floor{\frac{na}{b}} = \frac{(a - 1)(b - 1)}{2}.$$
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When $b = 1$, the sum on the left is understood to be $0$.
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\subsection*{\unverified{Exercise 7a}}%
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\hyperlabel{sub:exercise-7a}%
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\label{sub:exercise-7a}
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Derive this result by a geometric argument, counting lattice points in a right
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triangle.
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@ -212,7 +212,7 @@ Derive this result by a geometric argument, counting lattice points in a right
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\end{proof}
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\subsection*{\proceeding{Exercise 7b}}%
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\hyperlabel{sub:exercise-7b}%
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\label{sub:exercise-7b}
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Derive the result analytically as follows:
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By changing the index of summation, note that
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@ -226,7 +226,7 @@ Now apply Exercises 4(a) and (b) to the bracket on the right.
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\end{proof}
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\section*{\unverified{Exercise 8}}%
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\hyperlabel{sec:exercise-8}%
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\label{sec:exercise-8}
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Let $S$ be a set of points on the real line.
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The \textit{characteristic function} of $S$ is, by definition, the function
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@ -11,7 +11,7 @@
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\header{A Set of Axioms for the Real-Number System}{Tom M. Apostol}
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\section*{\verified{Lemma 1}}%
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\hyperlabel{sec:lemma-1}%
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\label{sec: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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@ -31,7 +31,7 @@ Nonempty set $S$ has supremum $L$ if and only if set $-S$ has infimum $-L$.
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\end{proof}
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\section*{\verified{Theorem I.27}}%
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\hyperlabel{sec:theorem-i.27}%
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\label{sec: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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is, there is a real number $L$ such that $L = \inf{S}$.
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@ -51,7 +51,7 @@ Every nonempty set $S$ that is bounded below has a greatest lower bound; that
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\end{proof}
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\section*{\verified{Theorem I.29}}%
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\hyperlabel{sec:theorem-i.29}
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\label{sec: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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@ -71,7 +71,7 @@ For every real $x$ there exists a positive integer $n$ such that $n > x$.
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\end{proof}
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\section*{\verified{Theorem I.30}}%
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\hyperlabel{sec:theorem-i.30}%
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\label{sec: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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integer $n$ such that $nx > y$.
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@ -92,7 +92,7 @@ If $x > 0$ and if $y$ is an arbitrary real number, there exists a positive
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\end{proof}
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\section*{\verified{Theorem I.31}}%
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\hyperlabel{sec:theorem-i.31}%
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\label{sec:theorem-i.31}
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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 $x = a$.
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@ -135,7 +135,7 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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\end{proof}
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\section*{\verified{Lemma 2}}%
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\hyperlabel{sec:lemma-2}%
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\label{sec:lemma-2}
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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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@ -178,12 +178,12 @@ If three real numbers $a$, $x$, and $y$ satisfy the inequalities
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\end{proof}
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\section*{Theorem I.32}%
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\hyperlabel{sec:theorem-i.32}%
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\label{sec:theorem-i.32}
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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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\hyperlabel{sub:theorem-i.32a}%
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\label{sub: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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@ -205,7 +205,7 @@ If $S$ has a supremum, then for some $x$ in $S$ we have $x > \sup{S} - h$.
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\end{proof}
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\subsection*{\verified{Theorem I.32b}}%
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\hyperlabel{sub:theorem-i.32b}%
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\label{sub: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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@ -227,7 +227,7 @@ If $S$ has an infimum, then for some $x$ in $S$ we have $x < \inf{S} + h$.
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\end{proof}
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\section*{Theorem I.33}%
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\hyperlabel{sec:theorem-i.33}%
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\label{sec:theorem-i.33}
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Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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$$C = \{a + b : a \in A, b \in B\}.$$
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@ -235,7 +235,7 @@ Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
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\note{This is known as the "Additive Property."}
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\subsection*{\verified{Theorem I.33a}}%
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\hyperlabel{sub:theorem-i.33a}%
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\label{sub: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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$$\sup{C} = \sup{A} + \sup{B}.$$
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@ -250,7 +250,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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$\sup{A} + \sup{B}$ is the \textit{least} upper bound of $C$.
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\paragraph{(i)}%
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\hyperlabel{par:theorem-i.33a-i}%
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\label{par:theorem-i.33a-i}
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Let $x \in C$.
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By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
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@ -303,7 +303,7 @@ If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
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\end{proof}
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\subsection*{\verified{Theorem I.33b}}%
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\hyperlabel{sub:theorem-i.33b}%
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\label{sub: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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$$\inf{C} = \inf{A} + \inf{B}.$$
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@ -318,7 +318,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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$\inf{A} + \inf{B}$ is the \textit{greatest} lower bound of $C$.
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\paragraph{(i)}%
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\hyperlabel{par:theorem-i.33b-i}%
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\label{par:theorem-i.33b-i}
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Let $x \in C$.
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By definition of $C$, there exist elements $a' \in A$ and $b' \in B$ such
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@ -371,7 +371,7 @@ If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
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\end{proof}
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\section*{\verified{Theorem I.34}}%
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\hyperlabel{sec:theorem-i.34}%
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\label{sec: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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for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
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@ -11,7 +11,7 @@
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\header{Useful Facts About Sets}{Herbert B. Enderton}
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\section*{\proceeding{Lemma 0A}}%
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\hyperlabel{sec:lemma-0a}%
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\label{sec:lemma-0a}
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Assume that $\langle x_1, \ldots, x_m \rangle =
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\langle y_1, \ldots, y_m, \ldots, y_{m+k} \rangle$.
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@ -13,7 +13,7 @@ We assume there exists a class $\mathscr{M}$ of measurable sets in the plane and
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properties:
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\section*{\defined{Nonnegative Property}}%
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\hyperlabel{sec:nonnegative-property}%
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\label{sec:nonnegative-property}
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For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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@ -24,7 +24,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
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\end{axiom}
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\section*{\defined{Additive Property}}%
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\hyperlabel{sec:additive-property}%
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\label{sec:additive-property}
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If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
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$\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
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@ -36,7 +36,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
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\end{axiom}
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\section*{\defined{Difference Property}}%
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\hyperlabel{sec:difference-property}%
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\label{sec:difference-property}
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If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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$\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
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@ -48,7 +48,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
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\end{axiom}
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\section*{\defined{Invariance Under Congruence}}%
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\hyperlabel{sec:invariance-under-congruence}%
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\label{sec:invariance-under-congruence}
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If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
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also in $\mathscr{M}$ and we have $a(S) = a(T)$.
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@ -72,7 +72,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
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\end{axiom}
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\section*{\proceeding{Exhaustion Property}}%
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\hyperlabel{sec:exhaustion-property}%
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\label{sec:exhaustion-property}
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Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
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that
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@ -9,7 +9,7 @@
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\begin{document}
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\section{\proceeding{Sum of Arithmetic Series}}%
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\hyperlabel{sec:sum-arithmetic-series}%
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\label{sec:sum-arithmetic-series}
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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Then for some $n \in \mathbb{N}$,
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|
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@ -9,7 +9,7 @@
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\begin{document}
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\section{\proceeding{Sum of Geometric Series}}%
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\hyperlabel{sec:sum-geometric-series}%
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||||
\label{sec:sum-geometric-series}
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|
||||
Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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Then for some $n \in \mathbb{N}$,
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||||
|
|
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|
@ -16,9 +16,6 @@
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|||
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\hypersetup{colorlinks=true, urlcolor=blue}
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||||
\newcommand{\leanref}[2]{\color{blue}$\pmb{\exists}\;{-}\;$\href{#1}{#2}}
|
||||
\newcommand{\hyperlabel}[1]{%
|
||||
\label{#1}%
|
||||
\hypertarget{#1}{}}
|
||||
|
||||
% ========================================
|
||||
% Environments
|
||||
|
|
|
|||
Loading…
Reference in New Issue