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 Friday, December 1st (Sections VI.1, VI.2, VI.3, and VI.4)
- 1. Show directly from the definition of the Riemann integral that $\int_0^2 x\ dx = 2$. [Hint: given a partition of $[0,2]$, break the triangle up into suitable trapezoids.]
- 2. 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 $\int_0^1 f(x)\ dx = 0$.
- 3. 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 $\int_0^1 f(x)\ dx=0$.
- 4. 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 $\int_a^b f(x)\ dx > 0$. [Note: you may assume, without proof, that that $f$ is integrable. We will show this to be the case on Friday, December 1st using the fact that $f$ is continuous.]
- 5. Suppose $f\colon[a,b]\to\mathbb{R}$ is continuous and $f(x)\geq 0$ for all $x\in[a,b]$. Show that \[ \lim_{n\to\infty} \left(\int_a^b f(x)^n\ dx \right)^{1/n}=\max_{a\leq x\leq b} f(x). \] [Hint: you may use, without proof, the fact that $\displaystyle \lim_{n\to\infty} r^{1/n}=1$ for any $r > 0$.]
Homework 11, due Monday, November 20th (Sections V.1, V.2, and V.3)
- 1. Fix $n\in \mathbb{N}$ and let $f\colon \mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^n$. Using the $\epsilon$-$\delta$ definition, show that $f$ is differentiable on $\mathbb{R}$.
- 2. For $U\subset\mathbb{R}$ an open interval, let $f\colon U\to\mathbb{R}$ be differentiable. Suppose there exists $R > 0$ so that $|f'(x)|\leq R$ for all $x\in U$. Show that $f$ is uniformly continuous.
- 3. For $U\subset\mathbb{R}$ open, let $f\colon U\to\mathbb{R}$ be twice differentiable. Suppose there exists $x_0\in U$ such that $f'(x_0)=0$ and $f''(x_0) < 0$. Show that there exists $\delta > 0$ so that
\[
f(x_0) = \max\{ f(x)\colon x_0-\delta < x < x_0+\delta\};
\]
that is, $f$ attains a local maximum at $x_0$.
[Remark: for $f''(x_0) > 0$ this holds after replacing "max" with "min," in which case we say $f$ attains a local minimum.] - 4. [Optional Exercise: L'Hôpital's Rule] For $a,b\in \mathbb{R}$ with $a < b$, let $f,g\colon (a,b)\to\mathbb{R}$ be differentiable functions.
- (a) Show that there exists $c\in (a,b)$ such that \[ f'(c)(g(b) - g(a)) = g'(c) ( f(b) - f(a)). \] [Hint: consider $F(x) = (f(x) - f(a))(g(b) - g(a)) - (f(b) - f(a)) (g(x) - g(a))$.]
- (b) Assume that $g$ and $g'$ are non-zero on $(a,b)$ and that $\displaystyle \lim_{x\to a} f(x) = \lim_{x\to a} g(x) =0$. Show that \[ \lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}, \] provided the limit on the right exists.
Homework 10, due Wednesday, November 15th (Section IV.6)
- 1. For each $n\in \mathbb{N}$, define $f_n\colon \mathbb{R}\to \mathbb{R}$ by $f_n(x)= n\left[ \left(x+\frac1n\right)^3 - x^3\right]$. Show that $(f_n)_{n\in\mathbb{N}}$ converges pointwise on $\mathbb{R}$ to $f(x)=3 x^2$.
- 2. For each $n\in \mathbb{N}$, define $f_n\colon \mathbb{R}\to \mathbb{R}$ by $f_n(x) = 1 + x + x^2 + \cdots +x^n$, and define $f\colon \mathbb{R}\setminus\{1\}\to\mathbb{R}$ by $\displaystyle f(x) = \frac{1}{1-x}$.
- (a) Show that $(f_n)_{n\in \mathbb{N}}$ converges pointwise on $(-1,1)$ to $f$.
- (b) Show that $(f_n)_{n\in\mathbb{N}}$ does not converge at any $x\not\in (-1,1)$.
- 3. Let $(f_n)_{n\in \mathbb{N}}$ and $f$ be as in the previous exercise.
- (a) Show that $(f_n)_{n\in \mathbb{N}}$ does not converge uniformly on $(-1,1)$ to $f$.
- (b) Show that for any $0 < r < 1$, $(f_n)_{n\in \mathbb{N}}$ converges uniformly on $[-r,r]$ to $f$.
- 4. For each $n\in \mathbb{N}$ define $f_n\colon \mathbb{R}\to\mathbb{R}$ by $\displaystyle f_n(x) = \frac{nx}{1+nx^2}$. Show that $(f_n)_{n\in\mathbb{N}}$ converges pointwise on $\mathbb{R}$, but not uniformly on $\mathbb{R}$. [Hint: for the latter, consider the continuity of the limit function.]
- 5. 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 (where $C(E',\mathbb{R})$ and $C(E,\mathbb{R})$ are both given the metric $D$ from class).
- (b) Show that if $\varphi$ is onto, then $D(\Phi(f), \Phi(g)) = D(f,g)$ for all $f,g\in C(E',\mathbb{R})$.
Homework 9, due Wednesday, November 8th (Sections IV.4 and IV.5)
- 1. 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.
- 2. Let $(E,d)$ be a metric space, and let $f,g\colon E\to \mathbb{R}$ be bounded, uniformly continuous functions, where $\mathbb{R}$ is equipped with the usual metric. Show that the product $f\cdot g\colon E\to \mathbb{R}$ is bounded and uniformly continuous.
- 3. Equip the interval $(0,1)\subset \mathbb{R}$ with the usual metric.
- (a) Show that if $f\colon (0,1)\to \mathbb{R}$ is uniformly continuous, then it is bounded.
- (b) Give an example of a function $f\colon (0,1)\to\mathbb{R}$ that is continuous but unbounded.
- 4. Let $(E,d)$ be a metric space. Show that $E$ is disconnected if and only if there exists a continuous onto function $f\colon E \to \{0,1\}$.
- 5. For $f\colon \mathbb{R}\to\mathbb{R}$ an odd degree polynomial, show $f(\mathbb{R})=\mathbb{R}$.
Homework 8, due Wednesday, October 25th (Sections III.6 and IV.1)
- 1. Let $(E,d)$ be a metric space. Show that any finite set $S$ with at least two elements is disconnected.
- 2. Let $(E,d)$ be a metric space. Suppose $f\colon [0,1]\to E$ and $c>0$ are such that \[ d(f(s), f(t) ) = c |s-t|\qquad \forall s,t\in [0,1]. \] Show that the set $S=f([0,1])\subset E$ is connected.
- 3. 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.
- 4. Let $(E,d)$ be a metric space and fix $y\in E$. Show that the map $f\colon E\to \mathbb{R}$ defined by $f(x)=d(x,y)$ is continuous.
- 5. Consider the function $f\colon \mathbb{R}\to\mathbb{R}$ defined by \[ f(x)=\begin{cases} \frac{1}{m} & x\in\mathbb{Q} \text{ with $x=\frac{n}{m}$ for $n\in \mathbb{Z}$ and $m\in \mathbb{N}$ sharing no common factors}\\ 0 & x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}. \] Show that $f$ is continuous at every $x\in \mathbb{R}\setminus\mathbb{Q}$, but discontinuous at every $x\in \mathbb{Q}$.
Homework 7, due Wednesday, October 18th (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 $(x_n)_{n\in \mathbb{N}}\subset [0,1]$. In this exercise we give a direct proof that there is a convergent subsequence.
- (a) Show that there is sequence of closed intervals \[ [0,1]\supset I_1\supset I_2\supset \cdots \] such that each $I_k$ has length $2^{-k}$ and contains infinitely many of the $x_n$.
- (b) Find a subsequence $(x_{n_k})_{k\in \mathbb{N}}$ satisfying $x_{n_k}\in I_k$ for each $k\in \mathbb{N}$, and show that it converges.
- 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 $\epsilon$. Show that every sequentially compact metric space is totally bounded and complete. (One can further show a set is compact iff it is sequentially compact iff it is totally bounded and complete.)
- 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 $\{S_i\}_{i\in I}$ be a collection of connected subsets. Show that if $\displaystyle \bigcap_{i\in I} S_i \neq\emptyset$, then $\displaystyle \bigcup_{i\in I} S_i$ is connected.
Homework 6, due Wednesday, October 11th (Sections III.4 and III.5)
- 1. Show that in $\mathbb{R}^2$ with the 2-dimensional Euclidean metric, the following sequence is Cauchy: \[ \left( (1- \frac{1}{n}, \frac{2}{n-0.5})\right)_{n\in\mathbb{N}}. \]
- 2. Show that in $\mathbb{R}^2$ with the 2-dimensional Euclidean metric, the following sequence is not Cauchy: \[ \left( (\frac1n, n+\frac1n)\right)_{n\in \mathbb{N}}. \]
- 3. In a metric space $(E,d)$, let $(a_n)_{n\in \mathbb{N}}\subset E$ be a convergent sequence with limit $a\in E$. Suppose $A,B\subset \mathbb{N}$ are infinite subsets such that $A\cap B=\emptyset$ and $A\cup B =\mathbb{N}$. For $b\in E\setminus\{a\}$ define a sequence $(x_n)_{n\in\mathbb{N}}\subset E$ by \[ x_n:=\begin{cases} x_n = a_n & \text{if }n\in A\\ x_n = b & \text{if }n\in B \end{cases}. \] Show that $(x_n)_{n\in \mathbb{N}}$ is not a Cauchy sequence.
- 4. On $\mathbb{R}^2$, consider the Jungle River metric: \[ d( (x,y), (x',y')) = \begin{cases} |y| + |y'| + |x-x'| & \text{if }x\neq x'\\ |y-y'| & \text{if }x=x' \end{cases}. \] Show that $(\mathbb{R}^2,d)$ is complete. [Hint: once you have determined a possible limit $(x,y)$, consider the cases $y=0$ and $y\neq 0$ separately. It may also be beneficial to draw what the open balls look like in this metric for each of these cases.]
- 5. In $\mathbb{R}$ with the usual metric, consider the collection
\[
\mathscr{C}=\{B(x,1)\}_{x\in \mathbb{R}\setminus\mathbb{Q}}.
\]
- (a) Show that $\mathscr{C}$ is an open cover of $\mathbb{R}$.
- (b) Show that $\mathscr{C}$ has no finite subcovers of $\mathbb{R}$.
Homework 5, due Wednesday, October 4th (Section III.3)
- 1. Let $\sigma\colon \mathbb{N}\to\mathbb{N}$ be one-to-one and onto ($\sigma$ is called a permutation of $\mathbb{N}$). For $(E,d)$ a metric space, show that if $(x_n)_{n\in\mathbb{N}}\subset E$ converges to $x\in E$, 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.)
- 2. Prove that the following sequences in $\mathbb{R}$ (with the usual metric) converge. In particular, find their limits.
- (a) $\displaystyle x_n=\frac{3n}{n+1}$ for $n\in\mathbb{N}$.
- (b) $\displaystyle x_n=\frac{5n+3}{n^2-n+1}$ for $n\in\mathbb{N}$.
- (c) $\displaystyle x_n=\sqrt{n+1}-\sqrt{n}$ for $n\in\mathbb{N}$.
- 3. We say a sequence $(x_n)_{n\in \mathbb{N}}\subset \mathbb{R}$ (with the usual metric) diverges to $+\infty$ (resp. $-\infty$) if $\forall R>0$, $\exists N\in \mathbb{N}$ such that $\forall n\geq N$ \[ x_n \geq R\qquad \text{(resp. $x_n\leq -R$)} \] Show that $(x_n)_{n\in\mathbb{N}} \subset \mathbb{R}\setminus\{0\}$ diverges to $+\infty$ iff $\{x_n\colon N\in \mathbb{N}\}$ is bounded below and $\left( \frac{1}{x_n}\right)_{n\in \mathbb{N}}$ converges to zero.
- 4. Let $(x_n)_{n\in \mathbb{N}}\subset \mathbb{R}$ be a bounded sequence. Consider the sequences $(a_k)_{k\in \mathbb{N}}, (b_k)_{k\in \mathbb{N}}\subset \mathbb{R}$ defined by
\begin{align*}
a_k:=\sup\{x_n\colon n\geq k\}\qquad\text{and}\qquad b_k:=\inf\{x_n \colon n\geq k\}
\end{align*}
The limit supremum and limit infimum of $(x_n)_{n\in \mathbb{N}}$ are defined as
\begin{align*}
\limsup_{n\to\infty} x_n := \lim_{k\to\infty} a_k\qquad\text{and}\qquad\liminf_{n\to\infty} x_n := \lim_{k\to\infty} b_k,
\end{align*}
respectively.
- (a) Show that the limit supremum and limit infimum of $(x_n)_{n\in \mathbb{N}}$ always exists, and that \[ \limsup_{n\to\infty} x_n = \inf\{ a_k\colon k\in \mathbb{N}\}\qquad\text{and}\qquad \liminf_{n\to\infty} x_n = \sup\{ b_k\colon k\in \mathbb{N}\}. \]
(b) Show that
\[
\liminf_{n\to\infty} x_n \leq \limsup_{n\to\infty} x_n.
\]
- (c) Show that \[ \liminf_{n\to\infty} x_n = \limsup_{n\to\infty} x_n \] if and only if $(x_n)_{n\in \mathbb{N}}$ converges. Note that in this case \[ \lim_{n\to\infty} x_n = \liminf_{n\to\infty} x_n = \limsup_{n\to\infty} x_n. \]
Homework 4, due Wednesday, September 20th (Section III.2)
- 1. 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 4 and 5 on Homework 3, show that $d_1$, $d_2$, and $d_\infty$ are all equivalent to each other.
- 2. Show that the set \[ S:=\{(x,y)\in\mathbb{R}^2\colon y> x\} \] is open in the metric space $(\mathbb{R}^2,d_2)$, where $d_2$ is as in Exercise 1. [Hint: use Exercise 1.]
- 3. Show that the set \[ S:=\{\frac{1}{n}\colon n\in \mathbb{N}\} \] is not closed in $\mathbb{R}$ with the usual metric (i.e. $d(x,y)=|x-y|$).
- 4. Let $(E,d)$ be a metric space. For $x\in E$, show that the singleton set $\{x\}\subset E$ is always closed. Then, use this to show that for any finite subset $F\subset E$, $F$ is closed.
- 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 3, due Wednesday, September 13th (Sections II.3 and III.1)
- 1. Let $f\colon\mathbb{R}\to\mathbb{R}$ be the function defined by $f(x) = \frac{x}{\sqrt{1+x^2}}$.
- (a) Show that $\sup\{ f(x)\colon x\in \mathbb{R}\}=1$.
- (b) Show that $\inf\{f(x)\colon x\in \mathbb{R}\} = -1$.
- 2. Recall our definition of infinite decimal expansions from class.
- (a) Show that $\inf\left\{\frac{1}{10^n}\colon n\in \mathbb{N}\right\} =0$.
- (b) Consider the following infinite decimal expansion: \[ 0.999\ldots = \sup\{0.\underbrace{9\cdots 9}_{n \text{ digits}}\colon n\in \mathbb{N}\}. \] Prove that $0.999\ldots = 1$. [Hint: give yourself and $\epsilon$ of room.]
- 3. Suppose $a_1,a_2,\ldots, a_n\in \mathbb{R}$ satisfy $a_1+a_2+\cdots +a_n\geq 1$. Prove that \[ a_1^2+a_2^2+\cdots + a_n^2 \geq \frac{1}{n}. \]
- 4. 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 $(0,0)$ and radius $1$.
- 5. 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 $(0,0)$ and radius $1$.
Homework 2, due Wednesday, September 6th (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 $a,b\in \mathbb{R}$ with $a < b$ there exists $s\in S$ with $a < s < b$.
- (b) Show that the set of irrational numbers $\mathbb{R}\setminus\mathbb{Q}$ is dense.
Homework 1, due Wednesday, August 30th (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. Let $x,y\in\mathbb{R}$ and $\epsilon\in\mathbb{R}_+$.
- (a) Show that $|x| < \epsilon$ iff $-\epsilon < x < \epsilon$. [Hint: first show that $a < b$ implies $-a > -b$ for $a,b\in\mathbb{R}$.]
- (b) Show that $|x-y| < \epsilon$ iff $y - \epsilon < x < y + \epsilon$.
- 5. For $x,y\in\mathbb{R}$, show that if $|x-y|<\epsilon$ for every $\epsilon\in\mathbb{R}_+$, then $x=y$.
