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## 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. 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. 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. 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). 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. 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. 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. 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. 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. 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.