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.

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

3. Do Exercise 1.3.9.

Solution sketches.

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.

1. Do Exercise 2.2.4.

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

3. Do Exercise 2.4.4(a).

Solution sketches.

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.

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.

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.

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.

1. Do Exercise 4.5.3.

2. Do Exercise 4.5.4.

3. Do Exercise 4.5.7.

Solution sketches.

1. Do Exercise 5.2.2.

2. Do Exercise 5.2.9.

3. Do Exercise 5.3.2.

Solution sketches.

1. Do Exercise 6.3.1.

2. Do Exercise 6.4.7(a).

3. Do Exercise 7.2.1.

Solution sketches.