# 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

- 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:

- $\forall \text{ rational } \epsilon>0 \exists N\in \mathbb{N} (n>N\rightarrow a_n>b_n-\epsilon)$
- $(a_n)\sim (b_n) \text{ OR } \exists N (n>N\rightarrow a_n>b_n)$

*Show that $[(a_n)]<[(b_n)]$ if and only if $\exists \text{ rational }\epsilon>0 \exists N (n>N\rightarrow b_n>a_n+\epsilon)$
- Due 2/13: Chapter 2:17*,18,19,21,22,23,26,30,31*
- Due 2/20: Chapter 1:39, Chapter 2:1,2,3,5,11,12,14,34
- Due 3/4: Chapter 2:6,81,83a,84,92,130
- Due: 3/13: Chapter 2:40*,41,43,71. Prelim Problems (see page 135): 5*,13,14*
- Due 3/22: Chapter 2:54,55,58,59,68,78*. Prelim Problem:15*
- 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
- 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 $\omega_2\preceq \omega_1$.)
- 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.
## Instructions from the grader:

Recall that this is a proof-based course, and thus a strict level of rigor is necessary in your solutions. Pictures are never proofs, but are always encouraged along with proofs.

Please put the problems in order and staple your homework in the upper left corner. UNSTAPLED HOMEWORK will be creatively dealt with at the discretion of the grader. Please use complete sentences. Your solution should not be just a sequence of equations and formulas. Make sure that your presentation is clean. Do not submit solutions with crossed out parts. Recopy problems if necessary.

It is okay, even encouraged, if you work on the homeworks in groups. The written solutions, though, must be your own.