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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:[a,b]R be integrable on [a,b]. Show that |f| is also integrable on [a,b] with |baf(x) dx|ba|f(x)| dx.
  • 2. Let f:[a,b]R be a continuous function such that f(x)0 for all x[a,b]. Show that if there exists x0[a,b] such that f(x0)>0, then baf(x) dx>0.
  • 3. Suppose f:[a,b]R is continuous and f(x)0 for all x[a,b]. Show that lim
  • 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 12 Solutions
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 11 Solutions
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.
    [Hint: you may quote the result from Exercise 8 in Chapter V of Rosenlicht.]
  • 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 10 Solutions (Exercise 2 was graded, the remaining exercises were checked for completion.)
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 9 Solutions (Exercise 1 was graded, the remaining exercises were checked for completion.)
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 8 Solutions (Exercise 2 was graded, the remaining exercises were checked for completion.)
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 7 Solutions (Exercise 2 was graded, the remaining exercises were checked for completion.)
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 6 Solutions (Exercise 4 was graded, the remaining exercises were checked for completion.)
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 5 Solutions (Exercise 2 was graded, the remaining exercises were checked for completion.)
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 4 Solutions (Exercises 3.(a) and 5 were graded.)
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 3 Solutions (Exercise 4 was graded.)
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 2 Solutions (Exercises 1 and 3 were graded.)
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.
Homework 1 Solutions (Exercises 4 and 5 were graded.)

Midterm Exams

Midterm 1 is in class on Thursday, September 29th. Here is a list of relevant textbook questions: Chapter I Exercises 2, 3, 4, 5; Chapter II Exercises 5, 6, 7, 10, 12, 13, 14; Chapter III Exercises 1, 3, 4, 6, 8, 9, 10, 16, 17, 18. Solutions.

Midterm 2 is in class on Thursday, November 3rd. Here is a list of relevant textbook questions: Chapter III Exercises 24-38; Chapter IV Exercises 1-4, 6-11, 13-21, 24-30, 32-41. Solutions.

Final Exam

The Final exam is on Wednesday, December 14th from 8:00 am to 11:00 am in Cory 289. Here is a list or relevant textbook questions to supplement those suggested for the midterm exams: Chapter IV Exercises 42-44, 46; Chapter V Exercises 1-4, 6-15; Chapter VI Exercises 1-5, 7-28; Chapter VII Exercises 6-15.

Extra Credit Assignment

The Completion of a Metric Space. Due by 11:00 am on Friday, November 4th. Solutions.