Homework Assignments
Homework is collected at the beginning of lecture of the specified day. No late homework will be accepted. Please write your name and the homework number on each assignment. If you would like to try writing your homework with LaTeX, here is a template that produces this output.
Homework 12, due Thursday, December 1st (Sections VI.3, VI.4, and VI.5)
- 1. Let $f\colon[a,b]\to\mathbb{R}$ be integrable on $[a,b]$. Show that $|f|$ is also integrable on $[a,b]$ with \[ \left|\int_a^b f(x)\ dx\right| \leq \int_a^b |f(x)|\ dx. \]
- 2. Let $f\colon[a,b]\to\mathbb{R}$ be a continuous function such that $f(x)\geq 0$ for all $x\in [a,b]$. Show that if there exists $x_0\in [a,b]$ such that $f(x_0)>0$, then $\int_a^b f(x)\ dx>0$.
- 3. Suppose $f\colon[a,b]\to\mathbb{R}$ is continuous and $f(x)\geq 0$ for all $x\in[a,b]$. Show that \[ \lim_{n\to\infty} \left(\int_a^b f(x)^n\ dx \right)^{1/n}=\max_{a\leq x\leq b} f(x). \]
- 4. Define $f\colon \mathbb{R}\to\mathbb{R}$ by \[ f(x)=\begin{cases} 0 & \text{if }x\leq 0\\ e^{-\frac{1}{x^2}} & \text{if }x>0\end{cases}. \] Show that $f^{(n)}(x)$ exists for all $x\in \mathbb{R}$ and $n\in\mathbb{N}$. In particular, show that $f^{(n)}(0)=0$ for each $n\in \mathbb{N}$.
- 5. For $a,b\in \mathbb{R}$ with $a < b$, define $\psi_{[a,b]}\colon \mathbb{R}\to\mathbb{R}$ by
\[
\psi_{[a,b]}(x) = \begin{cases} \exp\left(-\frac{1}{(x-a)^2}-\frac{1}{(x-b)^2}\right) & \text{if }a < x < b\\
0 & \text{otherwise}\end{cases}.
\]
- (a) Show that $\psi_{[a,b]}^{(n)}(x)$ exists for all $x\in \mathbb{R}$ and $n\in \mathbb{N}$.
- (b) For each $n\in \mathbb{N}$, define $\displaystyle c_n:=\int_{-1}^1 \psi_{\left[-\frac1n,\frac1n\right]}(x)\ dx$. Show that $c_n>0$ for all $n\in\mathbb{N}$ but $\displaystyle \lim_{n\to\infty} c_n=0$.
- (c) Let $f\colon [a,b]\to\mathbb{R}$ be continuous. Show that for $x_0\in (a,b)$ \[ \lim_{n\to\infty} \int_{a}^b \frac{1}{c_n} \psi_{\left[-\frac1n,\frac1n\right]}(x_0-x) f(x)\ dx = f(x_0). \]
Homework 11, due Tuesday, November 22nd (Sections VI.1 and VI.2)
- 1. Consider the function $f\colon [0,1]\to \mathbb{R}$ defined by \[ f(x)=\begin{cases} 1 & \text{if }x=\frac{1}{n}\text{ for some }n\in \mathbb{N}\\ 0 & \text{otherwise}\end{cases} \] Show that $f$ is Riemann integrable on $[0,1]$.
- 2. Consider the function $f\colon [0,1]\to \mathbb{R}$ defined by \[ f(x)=\begin{cases} \frac{1}{m} & \text{if }x=\frac{n}{m}\text{ for $n,m\in \mathbb{N}$ with no common factors}\\ 0 & \text{otherwise}\end{cases}. \] Show that $f$ is Riemann integrable on $[0,1]$.
- 3. Show that the composition of two Riemann integrable functions need not be Riemann integrable. [Hint: there is a reason this is the last exercise.]
Homework 10, due Thursday, November 17th (Sections V.2, V.3, V.4, and VI.1)
- 1. [L'Hôpital's Rule] For $a,b\in \mathbb{R}$, $a < b$ suppose $f,g\colon (a,b)\to\mathbb{R}$ are differentiable. Assume $g$ and $g'$ are non-zero on $(a,b)$, and that $\displaystyle \lim_{x\to a^+} \frac{f'(x)}{g'(x)}$ exists. Show that
\[
\lim_{x\to a^+} \frac{f(x)}{g(x)} = \lim_{x\to a^+} \frac{f'(x)}{g'(x)}
\]
holds if:
- (a) $\displaystyle \lim_{x\to a^+} f(x) = \lim_{x\to a^+} g(x)=0$; or
- (b) [This exercise is no longer being collected.] $\displaystyle \lim_{x\to a^+} \frac{1}{f(x)} = \lim_{x\to a^+} \frac{1}{g(x)}=0$.
- 2. For each $n\in \mathbb{N}$, consider the function $f_n\colon \mathbb{R}\to\mathbb{R}$ defined by $f(x)=|x|^{1+\frac{1}{n}}$.
- (a) Show that $f_n$ is differentiable on $\mathbb{R}$ and that $f_n'$ is continuous.
- (b) Show that $(f_n)_{n\in\mathbb{N}}$ converges uniformly to $f(x)=|x|$ on $[-1,1]$.
- (c) Show that $(f_n')_{n\in \mathbb{N}}$ converges pointwise to \[ g(x)=\begin{cases} -1 & \text{if }x < 0\\ 0 & \text{if }x=0\\ 1 & \text{if }x > 0 \end{cases}. \]
- 3. For $a,b\in \mathbb{R}$, $a < b$ suppose $(f_n)_{n\in \mathbb{N}}$ is a sequence of differentiable functions $f_n\colon (a,b)\to\mathbb{R}$. Assume that $(f_n)_{n\in\mathbb{N}}$ converges uniformly to some $f\colon (a,b)\to\mathbb{R}$ and that $(f_n')_{n\in\mathbb{N}}$ converge uniformly to some $g\colon (a,b)\to\mathbb{R}$. Show that $f$ is differentiable on $(a,b)$ with $f'=g$.
[Remark: contrast this with the previous exercise.] - 4. Let $U\subset \mathbb{R}$ be open, and let $f\colon U\to\mathbb{R}$ be twice differentiable at some $x_0\in U$. Show that if $f'(x_0)=0$ and $f''(x_0) < 0$, then there exists $\delta > 0$ so that $f$ attains a maximum on $B(x_0,\delta)$ at $x_0$.
[Remark: in order for $f''(x_0)$ to be defined, it is implicitely assumed that $f'(x)$ exists for $x\in B(x_0,r)$ for some $r>0$. If $f''(x_0) > 0$, then the statement is true after replacing "maximum" with "minimum."] - 5. [This exercise is no longer being collected.] For $a,b\in \mathbb{R}$, $a < b$ suppose $f\colon [a,b]\to\mathbb{R}$ is integrable. Show that $|f|$ is integrable on $[a,b]$ and that \[ \left| \int_a^b f(x)\ dx\right| \leq \int_a^b |f(x)|\ dx. \]
Homework 9, due Thursday, November 10th (Sections IV.6 and V.1)
- 1. Let $(E,d)$ and $(E',d')$ be metric spaces. A family $\{f_i\}_{i\in I}$ of functions $f_i\colon E\to E'$, $i\in I$, is said to be uniformly equicontinuous if for all $\epsilon > 0$ there exists $\delta > 0$ such that whenever $x_1,x_2\in E$ satisfy $d(x_1,x_2) < \delta$ then $d'(f_i(x_1),f_i(x_2)) < \epsilon$ for all $i\in I$. Assume that $E$ is compact, and recall that $C(E,E')$ is the set of continuous functions from $E$ to $E'$, which has the metric $\displaystyle D(f,g)=\max_{x\in E} d'(f(x),g(x))$. Show that if $(f_n)_{n\in \mathbb{N}}\subset C(E,E')$ converges with respect to $D$, then $\{f_n\}_{n\in \mathbb{N}}$ is uniformly equicontinuous.
- 2. Let $(E,d)$ and $(E',d')$ be compact metric spaces. Let $\varphi\colon E\to E'$ be continuous. Define $\Phi\colon C(E',\mathbb{R})\to C(E,\mathbb{R})$ by $\Phi(f) = f\circ \varphi$.
- (a) Show that $\Phi$ is uniformly continuous.
- (b) Show that $\Phi$ is an isometry if and only if $\varphi$ is onto.
- 3. For each of the following sequences $(f_n)_{n\in\mathbb{N}}$, show whether or not they converge uniformly. If they do not converge uniformly, show whether or not they converge pointwise.
- (a) $f_n\colon \mathbb{R}\to\mathbb{R}$, $\displaystyle f_n(x)= \frac{x}{1+nx^2}$.
- (b) $f_n\colon \mathbb{R}\to\mathbb{R}$, $\displaystyle f_n(x) = \frac{nx}{1+nx^2}$.
- 4. Let $n\in \mathbb{N}$.
- (a) Show that $f\colon \mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^n$ is differentiable.
- (b) Show that $f\colon [0,\infty)\to\mathbb{R}$ defined by $f(x)=x^{1/n}$ is differentiable on $(0,\infty)$.
- 5. Let $(x_n)_{n\in\mathbb{N}}\subset\mathbb{R}$ be any strictly increasing sequence \[ x_1 < x_2 < x_3 < \cdots, \] which we assume converges to some $x_\infty\in \mathbb{R}$. Define $f\colon \mathbb{R}\to\mathbb{R}$ by \[ f(x)=\begin{cases} x_1 & \text{if }x\leq x_1\\\\ x_\infty - \frac{1}{n}(x_\infty - x_n) & \text{if }x_{n-1} < x\leq x_n\text{ for some }n\in\mathbb{N}\\\\ x_\infty & \text{if }x\geq x_\infty \end{cases}. \] Show that $f$ is differentiable at $x_\infty$, but $f$ is not continuous on $B(x_\infty, \delta)$ for any $\delta > 0$.
Homework 8, due Thursday, October 27th (Sections IV.4 and IV.5)
- 1. Let $\mathbb{R}$ have the usual metric, let $(E,d)$ be a metric space, and let $f\colon [0,\infty)\to E$ be a function. For $y_0\in E$, we say that $y_0$ is the limit of $f$ at infinity if for all $\epsilon > 0$ there exists $R>0$ such that if $x\in [0,\infty)$ satisfies $x> R$, then $d(f(x), y_0) < \epsilon$. We write $\displaystyle \lim_{x\to\infty} f(x)$ for $y_0$. (The idea here is that $R$ is a large number, and $x > R$ means $x$ is "close" to $\infty$, in which case we want $f(x)$ to be close to $y_0$.)
- (a) Let $(y_n)_{n\in\mathbb{N}}\subset E$ be a convergent sequence, say with limit $y_0\in E$. Define $f\colon[0,\infty)\to E$ by $f(x)=y_n$ when $n\in\mathbb{N}$ satisfies $n-1 < x\leq n$. Show that $\displaystyle \lim_{x\to\infty} f(x)=y_0$.
- (b) For $f\colon [0,\infty)\to E$, show that $\displaystyle \lim_{x\to\infty} f(x)$ exists if and only if $\displaystyle \lim_{x\to 0} f(x^{-1})$ exists, in which case these two limits are equal.
- 2. Let $n\in \mathbb{N}$.
- (a) Show that $f\colon \mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^n$ is not uniformly continuous for $n\geq 2$.
- (b) Show that $f\colon [0,\infty)\to\mathbb{R}$ defined by $f(x)=x^{1/n}$ is uniformly continuous.
- 3. Let $(E,d)$ be a metric space with a dense subset $S\subset E$, let $(E',d')$ be a complete metric space, and let $f\colon S\to E'$ be a uniformly continuous function. A continuous extension of $f$ to $E$ is a continuous function $g\colon E\to E'$ such that $g(x)=f(x)$ for all $x\in S$. Show that there is exactly one continuous extension of $f$ to $E$.
[Note: This means you have to show a continuous extension exists and that it is the only one.] - 4. Let $f\colon [a,b]\to\mathbb{R}$ be continuous and one-to-one. Show that $f([a,b])$ is either $[f(a),f(b)]$ or $[f(b),f(a)]$ (i.e. whichever makes sense as an interval).
- 5. Let $(E,d)$ be a metric space. A subset $S\subset E$ is path-connected if for any two points $x_1,x_2\in S$, there exists $a,b\in \mathbb{R}$ with $a < b$ and a continuous function $f\colon [a,b]\to S$ such that $f(a)=x_1$ and $f(b)=x_2$ (i.e. there is a "curve" or "path" starting at $x_1$ and ending at $x_2$ which stays inside $S$).
- (a) Show that a path-connected subset $S\subset E$ is connected.
- (b) In $n$-dimensional Euclidean metric space $(\mathbb{R}^n, d_n)$, show an open and connected subset $S\subset \mathbb{R}^n$ is path-connected.
[Hint: given $x_0\in S$, consider the set $A$ of points in $S$ which are connected to $x_0$ by a path and the set $B$ of points which are not connected to $x_0$ by a path.]
Homework 7, due Thursday, October 20th (Sections IV.1, IV.2, and IV.3)
- 1. Let $n\in \mathbb{N}$.
- (a) Show that $f\colon \mathbb{R}\to \mathbb{R}$ defined by $f(x)=x^n$ is continuous.
- (b) Show that $f\colon [0,\infty)\to\mathbb{R}$ defined by $f(x)=x^{1/n}$ is continuous.
- 2. Consider the function $f\colon \mathbb{R}\to\mathbb{R}$ defined by \[ f(x)=\begin{cases} \frac{1}{m} & x\in\mathbb{Q} \text{ with $x=\frac{n}{m}$ for $n\in \mathbb{Z}$ and $m\in \mathbb{N}$ with no common factors}\\ 0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}. \] Show that $f$ is continuous at every $x\in \mathbb{R}\setminus\mathbb{Q}$, but discontinuous at every $x\in \mathbb{Q}$.
- 3. Let $I_1,I_2\subset \mathbb{R}$ be intervals in $\mathbb{R}$ (not necessarily open or closed). Assume $f\colon I_1 \to I_2$ is onto and increasing: whenever $x,y\in I_1$ satisfy $x\leq y$ then $f(x)\leq f(y)$. Show that $f$ is continuous.
- 4. Find each limit and use the $\epsilon$-$\delta$-definition to show it is in fact the limit.
- (a) $\displaystyle \lim_{x\to 2} x^3$
- (b) $\displaystyle \lim_{x\to 1} \frac{x-1}{\sqrt{x}+1}$
- (c) $\displaystyle \lim_{(x_1,x_2)\to (0,0)} \frac{x_1^2x_2^2}{x_1^2+x_2^2}$ (where $\mathbb{R}^2$ has the $2$-dimensional Euclidean metric)
- 5. Let $(E,d)$ and $(E',d')$ be metric spaces. Let $f\colon E\to E'$ be a function and $x\in E$ a point. The oscillation of $f$ at $x$ is the quantity
\[
\omega_f(x):=\inf_{\delta > 0} \text{diam}(f(B(x,\delta))).
\]
- (a) Show that $f$ is continuous at $x_0$ if and only if $\omega_f(x_0)=0$.
- (b) For the function $f\colon \mathbb{R}\to \mathbb{R}$ defined by \[ f(x)=\begin{cases} 0 & \text{if }x\leq 0\\1 & \text{if }x > 0\end{cases}, \] show $\omega_f(0)=1$.
- (c) For the function $f\colon \mathbb{R}\to\mathbb{R}$ defined by \[ f(x)=\begin{cases} (-1)^n & \text{if }x=\frac{1}{n}\\ 0 & \text{otherwise}\end{cases}, \] show $\omega_f(0)=2$.
Homework 6, due Thursday, October 13th (Sections III.5 and III.6)
- 1. Let $(E,d)$ be a metric space.
- (a) Let $\{S_i\}_{i\in I}$ be a collection of compact subsets of $E$. Prove that their intersection $\displaystyle \bigcap_{i\in I} S_i$ is compact.
- (b) Let $S_1,S_2,\ldots, S_n\subset E$ be a finite number of compact subsets. Prove that their union $S_1\cup\cdots\cup S_n$ is also compact.
- 2. Let $A$ and $B$ be disjoint subsets of a metric space $(E,d)$. Suppose that $A$ is closed and $B$ is compact.
- (a) Show that \[ \inf\{d(a,b)\colon a\in A,\ b\in B\} > 0. \]
- (b) Give an example in $\mathbb{R}^2$ (with the $2$-dimensional Euclidean metric) that shows this does not hold when $B$ is assumed to be closed rather than compact.
- 3. Let $(E,d)$ be a metric space. $(E,d)$ is called sequentially compact if every sequence has a convergent subsequence. $(E,d)$ is called totally bounded if for every $\epsilon > 0$ the space $E$ can be covered by finitely many closed balls of radius $\epsilon$. In lecture we showed that every compact metric space is sequentially compact.
- (a) Show that every sequentially compact metric space is totally bounded and complete.
- (b) Show that every totally bounded and complete metric space is compact.
- 4. Let $(E,d)$ be a metric space and $S\subset E$ a subset.
- (a) Show that $A\subset S$ is open relative to $S$ if and only if $A=S\cap U$ for an open subset $U\subset E$.
- (b) Show that $B\subset S$ is closed relative to $S$ if and only if $B=S\cap V$ for a closed subset $V\subset E$.
- 5. Let $(E,d)$ be a metric space, and let $S\subset E$ be a connected subset. Suppose $T\subset E$ satisfies $S\subset T\subset \overline{S}$. Prove that $T$ is also connected.
- 6. Read Section 6 of Chapter III in Rosenlicht.
Homework 5, due Thursday, October 6th (Sections III.3 and III.4)
- 1. Let $(E,d)$ be a metric space and $(x_n)_{n\in\mathbb{N}}$ a sequence in $E$. We say $x\in E$ is an accumulation point of $(x_n)_{n\in\mathbb{N}}$ if for every $r>0$ there are infinitely many $n\in \mathbb{N}$ such that $x_n\in B(x,r)$. Show that $x\in E$ is an accumulation point of $(x_n)_{n\in\mathbb{N}}$ if and only if $\displaystyle \lim_{k\to\infty} x_{n_k}=x$ for some subsequence $(x_{n_k})_{k\in\mathbb{N}}$.
- 2. Let $(x_n)_{n\in\mathbb{N}}$ be a bounded sequence in $\mathbb{R}$ (equipped with the usual metric). Let $A\subset \mathbb{R}$ be the set of accumulation points of $(x_n)_{n\in\mathbb{N}}$. Prove \[ \limsup_{n\to\infty} x_n = \sup(A). \] [Remark: it is also true that $\displaystyle\liminf_{n\to\infty} x_n = \inf(A)$, but you do not need to prove this.]
- 3. Let $(E,d)$ be a metric space. For a non-empty subset $S\subset E$, the diameter of $S$ is the non-negative number \[ \text{diam}(S):=\sup\{d(x,y)\colon x,y\in S\}. \] Prove that $(E,d)$ is complete if and only if for any descending sequence $A_1\supset A_2\supset A_3\supset\cdots$ of non-empty closed sets with $\displaystyle \lim_{n\to\infty} \text{diam}(A_n) = 0$, the intersection $\displaystyle \bigcap_{n\in\mathbb{N}} A_n$ is non-empty.
- 4. Let $E$ be the set of bounded sequences of real numbers, equipped with the metric \[ d(\ (a_n)_{n\in \mathbb{N}},\ (b_n)_{n\in\mathbb{N}}):=\sup_{n\in \mathbb{N}} |a_n - b_n| \] Show that $(E,d)$ is complete. (You do not need to show that $d$ defines a metric.) [Hint: a convenient notation for a sequence of sequences is $\left( (x_n^{(k)})_{n\in \mathbb{N}}\right)_{k\in \mathbb{N}}$, so that for each fixed $k\in \mathbb{N}$, $(x_n^{(k)})_{n\in \mathbb{N}}$ is a bounded sequence of real numbers.]
Homework 4, due Thursday, September 22th (Sections III.2 and III.3)
- 1. Let $A,B$ be subsets in a metric space $(E,d)$.
- (a) If $A\subset B$, show $\overline{A}\subset \overline{B}$ and $A^\circ\subset B^\circ$.
- (b) Show $\overline{A\cup B}=\overline{A}\cup\overline{B}$ and $(A\cap B)^\circ = A^\circ\cap B^\circ$.
- (c) In $E=\mathbb{R}$ with the usual metric, give examples showing $\overline{A\cap B}\neq \overline{A}\cap \overline{B}$ and $(A\cup B)^\circ \neq A^\circ \cup B^\circ$.
- 2. For a subset $S$ in a metric space $(E,d)$, prove
- (a) $\overline{S}=\{x\in E\colon B(x,r)\cap S\neq\emptyset\ \forall r > 0\}$; and
- (b) $\partial S=\{x\in E\colon B(x,r)\cap S\neq\emptyset \text{ and }B(x,r)\cap S^c\neq \emptyset\ \forall r > 0\}$.
- 3. Let $(E,d)$ be a metric space. For non-empty subsets $A,B\subset \mathcal{E}$ define
\[
D(A,B):=\max\{ \sup_{a\in A}\inf_{b\in B} d(a,b),\ \sup_{b\in B}\inf_{a\in A}d(a,b)\}.
\]
[Here we are using the following notation: for a map $f\colon S\to\mathbb{R}$
\[
\inf_{s\in S} f(s):=\inf\{f(s)\colon s\in S\}\qquad\text{ and }\qquad\sup_{s\in S} f(s):=\sup\{f(s)\colon s\in S\}
\]
provided the infimum and supremum exists.]
- (a) For non-empty subsets $A,B\subset E$, prove that $D(A,B)=0$ if and only if $\overline{A}=\overline{B}$.
- (b) Let $\mathcal{E}=\{S\subset E\colon S\text{ is non-empty, closed, and bounded}\}$. Prove $(\mathcal{E},D)$ is a metric space.
- (c) For $E=\mathbb{R}^2$ and $d$ the 2-dimensional Euclidean metric, compute $D(A,B)$ for $A=B[(0,0),1]$ and $B= B[(0,0),3]\setminus B((0,0),2)$. [Hint: polar coordinates may prove useful.]
- 4. Prove that the following sequences in $\mathbb{R}$ (with the usual metric) converge. In particular, find their limits.
- (a) $x_n=\left(4+\frac{1}{n}\right)^2$ for $n\in \mathbb{N}$.
- (b) $x_n=\frac{3n}{n+1}$ for $n\in \mathbb{N}$.
- (c) $x_n=\frac{5n+3}{n^2+1}$ for $n\in \mathbb{N}$.
- (d) $x_n=\sqrt{n+1}-\sqrt{n}$ for $n\in \mathbb{N}$.
- 5. Let $\sigma\colon \mathbb{N}\to\mathbb{N}$ be one-to-one and onto ($\sigma$ is called a permutation of $\mathbb{N}$). Show that if $\{x_n\}_{n\in\mathbb{N}}$ converges to $x$, then $\{ x_{\sigma(n)}\}_{n\in\mathbb{N}}$ also converges to $x$. [This shows that any reordering of a convergent sequence still converges to the same point.]
Homework 3, due Thursday, September 15th (Sections III.1 and III.2)
- 1. Consider the map $d_1\colon \mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ defined for $\vec{x}=(x_1,x_2)$ and $\vec{y}=(y_1,y_2)$ in $\mathbb{R}^2$ as
\[
d_1(\vec{x},\vec{y}):=|x_1 - y_1|+|x_2 - y_2|
\]
- (a) Prove that $(\mathbb{R}^2, d_1)$ is a metric space.
- (b) In the Euclidean plane $\mathbb{R}^2$ and with respect to the metric $d_1$, draw a picture of $B[0,1]$, the closed ball with center $0$ and radius $1$.
- 2. Consider the map $d_\infty\colon \mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ defined for $\vec{x}=(x_1,x_2)$ and $\vec{y}=(y_1,y_2)$ in $\mathbb{R}^2$ as
\[
d_\infty(\vec{x},\vec{y}):=\max\{|x_1 - y_1|,|x_2 - y_2|\}
\]
- (a) Prove that $(\mathbb{R}^2, d_\infty)$ is a metric space.
- (b) In the Euclidean plane $\mathbb{R}^2$ and with respect to the metric $d_\infty$, draw a picture of $B[0,1]$, the closed ball with center $0$ and radius $1$.
- 3. Let $E$ be a set with two metrics $d_a$ and $d_b$. We say $d_a$ and $d_b$ are equivalent if there exists positive constants $c_1,c_2>0$ such that
\[
c_1 d_a(x,y)\leq d_b(x,y)\leq c_2 d_a (x,y)\qquad \forall x,y\in E
\]
- (a) For equivalent metrics $d_a$ and $d_b$ on $E$, prove that a subset $S\subset E$ is open in the metric space $(E,d_a)$ if and only if it is open in the metric space $(E,d_b)$.
- (b) Recall that the $2$-dimensional Euclidean metric on $\mathbb{R}^2$ is defined for $\vec{x}=(x_1,x_2)$ and $\vec{y}=(y_1,y_2)$ as \[ d_2(\vec{x},\vec{y}):=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}. \] Letting $d_1$ and $d_\infty$ be as in Exercises 1 and 2, show that $d_1$, $d_2$, and $d_\infty$ are all equivalent to each other.
- 4. Let $S\subset \mathbb{R}$ be non-empty, bounded, and open. Show that $S$ is a union of disjoint open intervals.
- 5. For $(E,d)$ a metric space, a point $x\in E$ is said to be isolated if the singleton set $\{x\}$ is open. Show that $x\in E$ is not isolated if and only if for every $r>0$ the set $B(x,r)$ has infinitely many elements.
Homework 2, due Thursday, September 8th (Sections II.3 and II.4)
- 1. Prove that if $S\subset\mathbb{R}$ is non-empty and bounded below, then it has an infimum.
- 2. For $S\subset\mathbb{R}$ a non-empty subset that is bounded above and $x\in \mathbb{R}$, let $xS$ be the set $\{xs\colon s\in S\}$.
- (a) Show that if $x>0$, then $\sup(xS)=x\sup(S)$.
- (b) Show that if $x<0$, then $\inf(xS)=x\sup(S)$.
- 3. Let $S,T\subset \mathbb{R}$ be non-empty subsets that are bounded from above, and define $S+T=\{s+t\colon s\in S,\ t\in T\}$. Show \[ \sup(S+T)=\sup(S)+\sup(T). \] Then, use this to prove that if $x\in \mathbb{R}$ and $S+x$ is the set $\{s+x\colon s\in S\}$, then $\sup(S+x)=\sup(S)+x$.
- 4. Recall that we say $S\subset\mathbb{R}$ is dense if for any $x\in\mathbb{R}$ and any $\epsilon > 0$ there exists $s\in S$ such that $|s-x| < \epsilon$.
- (a) Show that a set $S\subset\mathbb{R}$ is dense if and only if for any $a,b\in \mathbb{R}$ there exists $s\in S$ with $a < s < b$.
- (b) Show that the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ is dense.
- 5. Let $a_0\in \mathbb{N}\cup\{0\}$ and $\{a_1,a_2,\ldots\}\subset\{0,1,2,\ldots, 9\}$. Define the infinite decimal exansion $x=a_0.a_1a_2\cdots$ as we did in class. Show that for any $n\in\mathbb{N}$ we have \[ |x-a_0.a_1a_2\cdots a_n|\leq \frac{1}{10^n}. \] [Hint: use Exercise 3.]
Homework 1, due Thursday, September 1st (Sections II.1 and II.2)
- 1. Read Chapter I Notions from Set Theory in Rosenlicht.
- 2. Let $f\colon X\to Y$ be a function.
- (a) For a subset $A\subset X$, show $f^{-1}(f(A))\supset A$.
- (b) Show that $f$ is one-to-one iff $f^{-1}(f(A))=A$ for all $A\subset X$.
- (c) For a subset $B\subset Y$, show $f(f^{-1}(B))\subset B$.
- (d) Show that $f$ is onto iff $f(f^{-1}(B))=B$ for all $B\subset Y$.
- (e) For subsets $C,D\subset Y$, show $f^{-1}(C\cup D)=f^{-1}(C)\cup f^{-1}(D)$.
- (f) For subsets $C,D\subset Y$, show $f^{-1}(C\cap D)=f^{-1}(C)\cap f^{-1}(D)$.
- 3. For $a,b\in\mathbb{R}$ show
- (a) $\max\{a,b\}=\frac{1}{2}(a+b+|a-b|)$; and
- (b) $\min\{a,b\}=\frac{1}{2}(a+b-|a-b|)$.
- 4. For $x\in\mathbb{R}$ and $\epsilon\in\mathbb{R}_+$ show that $|x| < \epsilon$ iff $-\epsilon < x < \epsilon$. [Hint: first show that $a < b$ implies $-a > -b$ for $a,b\in\mathbb{R}$.]
- 5. For $x,y\in\mathbb{R}$, show that if $|x-y|<\epsilon$ for every $\epsilon\in\mathbb{R}_+$, then $x=y$.
