# Math 521

## Office hours

513 Van Vleck Hall
Wednesday 4:30-5:30
Friday 3:30-5:30 (3:30-4:30 preferred)

## Exams

There will be two in-class midterms on 2/22 and 4/5.

Solutions for the first exam: Solutions

Solutions for the second exam: Solutions

The first exam will cover chapter 1 (including our Cauchy sequence approach) and section 1 of chapter 2 (which ends on page 75).

## Homework

1. Due 2/4: Chapter 1: 1,2,14,15,16,*30, The operation $[(a_n)]\cdot [(b_n)]=[(a_n\cdot b_n)]$ is well-defined,
The following two definitions of $(a_n)\geq (b_n)$ are equivalent:
1. $\forall \text{ rational } \epsilon>0 \exists N\in \mathbb{N} (n>N\rightarrow a_n>b_n-\epsilon)$
2. $(a_n)\sim (b_n) \text{ OR } \exists N (n>N\rightarrow a_n>b_n)$

*Show that $[(a_n)]<[(b_n)]$ if and only if $\exists \text{ rational }\epsilon>0 \exists N (n>N\rightarrow b_n>a_n+\epsilon)$
2. Due 2/13: Chapter 2:17*,18,19,21,22,23,26,30,31*
3. Due 2/20: Chapter 1:39, Chapter 2:1,2,3,5,11,12,14,34
4. Due 3/4: Chapter 2:6,81,83a,84,92,130
5. Due: 3/13: Chapter 2:40*,41,43,71. Prelim Problems (see page 135): 5*,13,14*
6. Due 3/22: Chapter 2:54,55,58,59,68,78*. Prelim Problem:15*
7. Due 4/3: Show directly that every covering compact set is closed (this may be useful in the other problems), Chapter 2:46-53,94
8. Due 4/15:Chapter 3:7,11,13,14,26 (You should add to the list of assumptions in problem 26: For any $\omega_1,\omega_2$, either $\omega_1\preceq \omega_2$ or $\omega_2\preceq \omega_1$.)
9. Due: 5/6: Chapter 3:18,28-31,43,47,50,51
Homework solutions are available here

*Problems with a star are either harder or require some more serious detail-work. There will be several per homework assignment. For the honors option in the course, you must get more than 70% credit on the * problems.