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Math 3001002: Analysis 1, Fall 2018


Homework




Assignment

Assigned

Due

Problems

HW1 
9/6/18

9/12/18

Read Sections 1.11.3.
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 1.41.5.
1. Does the nonArchimedean 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 2.12.4.
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 2.52.7.
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, 3.13.2.
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
$$2\sin(k\theta)\sin(\theta/2)=\cos((k1/2)\theta)\cos((k+1/2)\theta).$$
(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, 3.33.4.
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 4.14.4.
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

11/28/18

12/5/18

Read Sections 6.16.4.
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.


