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Math 3001-002: Analysis 1, Fall 2018


Homework



Assignment
Assigned
Due
Problems
HW1 9/6/18
9/12/18
Read Sections 1.1-1.3.

1. Show that the coimage of a function is a partition of the domain of the function.

2. Do Exercise 1.2.13.

3. Do Exercise 1.3.2.

Solution sketches.

HW2 9/13/18
9/19/18
1. Do Exercise 1.3.3.

2. Do Exercise 1.3.6. (a), (b), (c).

3. Do Exercise 1.3.9.

Solution sketches.

HW3 9/20/18
9/26/18
Read Sections 1.4-1.5.

1. Does the non-Archimedean field $\mathbb R(t)$ satisfy the Nested Interval Property? Explain.

2. Show that if $S\subseteq [0,1]$ is uncountable, then there is a real number $r\in[0,1]$ such that both $[0,r]\cap S$ and $[r,1]\cap S$ are uncountable.

3. Do Exercise 1.5.8.

Solution sketches.

HW4 9/27/18
10/3/18
Read Sections 2.1-2.4.

1. Do Exercise 2.2.4.

2. Do Exercise 2.3.7 (a), (b), (c).

3. Do Exercise 2.4.4(a).

Solution sketches.

10/7/18

Read Sections 2.5-2.7.

Please read and think about the following exercises. (No need to write up the solutions.)

Exercises 2.4.2, 2.5.1, 2.5.2, 2.5.5, 2.5.8, 2.6.2, 2.6.3(a), 2.7.4.

HW5 10/18/18
10/24/18
Read Sections 2.8, 3.1-3.2.

1. Do Exercise 2.7.14(a).

2.
(a) Let $\theta$ be an arbitrary angle. Show that the partial sums of $\sum_{k=1}^{\infty} \sin(k\theta)$ are bounded. Hint: You may want to first prove and use the equality $$2\sin(k\theta)\sin(\theta/2)=\cos((k-1/2)\theta)-\cos((k+1/2)\theta).$$
(b) Show that $\sum_{k=1}^{\infty} \sin(k\theta)/k$ converges.

3. Do Exercise 3.2.3.

Solution sketches.

HW6 10/14/18
10/31/18
Read Sections 2.8, 3.3-3.4.

1. Do Exercise 3.3.4.

2. Do Exercise 3.3.6.

3. For each part below, give an example of a subset $A\subseteq \mathbb R$ or $A\subseteq \mathbb R^2$ such that
(a) $A$ is connected, but $A^{\circ}$ and $\partial A$ are not connected.
(b) $A$ is not connected, but $A^{\circ}$ and $\partial A$ are connected.

Solution sketches.

HW7 11/2/18
11/7/18
Read Sections 4.1-4.4. Read Exercises 4.3.11, 4.3.13, 4.4.11.

1. Do Exercise 4.3.6(a)(b)(c).

2. Show that a nonempty subset $C\subseteq \mathbb R$ is closed iff there is a continuous function $g:\mathbb R\to \mathbb R$ such that $C = g^{-1}(0)$.
(Hint for the proof of $\Leftarrow$: explain why the inverse image of a closed set is closed. Hint for the proof of $\Rightarrow$: you may cite parts of Exercise 4.3.12 if needed.)

3. A function $f:\mathbb R\to \mathbb R$ is periodic if there is a number $p$ such that $f(x+p) = f(x)$ for every $x$. (For example, $\sin(x)$ is periodic with $p=2\pi$, since $\sin(x+2\pi)=\sin(x)$.) Prove that a continuous periodic function is uniformly continuous.

Solution sketches.

HW8 11/7/18
11/14/18
Read Sections 4.5, 5.1, 5.2. Read Exercise 4.5.2(d).

1. Do Exercise 4.5.3.

2. Do Exercise 4.5.4.

3. Do Exercise 4.5.7.

Solution sketches.

HW9 11/15/18
11/28/18
Read Sections 5.3, 5.4, 5.5. Read Exercise 5.3.7.

1. Do Exercise 5.2.2.

2. Do Exercise 5.2.9.

3. Do Exercise 5.3.2.

Solution sketches.

HW10
Last one!
11/28/18
12/5/18
Read Sections 6.1-6.4. Read Exercise 6.5.3.

1. Do Exercise 6.3.1.

2. Do Exercise 6.4.7(a).

3. Do Exercise 7.2.1.

Solution sketches.