1. Do Exercise 1.2.4.

2. Do Exercise 1.2.7(b) and 1.2.7(c).

3. Do Exercise 1.2.8.

Solution sketches.

Read the following problems, but do not turn them in: 1.3.3, 1.3.6(a)(b)(c), 1.4.3, 1.4.8.

1. Do Exercise 1.3.2.

2. Do Exercise 1.3.9.

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

Solution sketches.

Read the following problems, but do not turn them in: 1.5.4, 1.5.5, 1.5.9, 1.6.10.

1. Do Exercise 1.5.2.

2. Do Exercise 1.5.8.

3. 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.

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.

1. Do Exercise 2.4.2. (Hint for (b): use MCT to prove $(y_n)$ converges.)

2. Do Exercise 2.4.3 (a). (Hint: use MCT to prove the sequence converges. To find the limit, show that it must satisfy $L^2-2=L$ and $L>0$.)

3. Do Exercise 2.5.1 (a), (b), (c).

Solution sketches.

1. Do Exercise 2.6.2 (a) (c).

2. Do Exercise 2.6.4 (c). (Where the author says ``Decide whether each of the following sequences is a Cauchy sequence'', interpret this to mean ``Decide whether each of the following sequences MUST BE a Cauchy sequence''.)

3. Do Exercise 2.7.4 (a) (d).

Solution sketches.

1. Do Exercise 2.7.14 (a). (You may assume Exercise 2.7.12.)

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

3.
(a) Let $\theta$ be an arbitrary angle. Show that $\sum_{k=1}^{\infty} \sin(k\theta)/k$ converges.
(b) Show that $\sum_{k=1}^{\infty} \sin(k\theta)/\sqrt{k}$ converges.

Solution sketches.

Read and think about Problems 3.2.4, 3.2.6, 3.2.7, 3.2.10, but do not turn them in.

1. Do Exercise 3.2.3.

2. Do Exercise 3.3.4.

3. Do Exercise 3.3.6. Explain why the statements are true for finite, give examples to show that they are false for closed, and then state (without proof or counterexample) whether they are true or false for compact.

Solution sketches.

Read and think about Problems 4.2.5, 4.2.6, 4.2.8(a)(b), 4.2.11, 4.3.4(a)(b), 4.3.5 but do not turn them in.

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

2. and 3. The goal is to 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 $C = g^{-1}(0)$.
For 2: Show the IF part. (Hint: explain why the inverse image of a closed set is closed.)
For 3: Show the ONLY IF part. (Hint: you may cite parts of Exercise 4.3.12 if needed.)

Solution sketches.

Read and think about Problems 4.4.8, 4.4.11, but do not turn them in.

1. Do Exercise 4.5.2.

2. Do Exercise 4.5.3.

3. Do Exercise 4.5.7. (Hint: Consider the cases (i) $f(0)=0$, (ii) $f(1)=1$, and (iii) both $f(0)>0$ and $f(1)<1$. In case (iii), make use of the function $f(x)-x$.)

Solution sketches.