Last updated Sat Oct 22 02:26 EDT 2016
Due-date: Friday 10/28/16.
You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. Don't make the mistake of thinking that I'm collecting only the problems I think are important.
The "due date" above is the date that your written-up problems should be handed in, but don't wait to get started on the assignment. You should always get started on problems as soon as we cover the relevant material in class.
- C: (Do this before parts A and B below.) Prove the following facts in the Interiors, Closures, and Boundaries handout: 4, 7–10, 16–22. To be prepared for the first exam, you should have these (as well as non-book B1–B3) done by Friday 10/14/16. Of these, hand in only #s 16 and 17.
- A: Rosenlicht pp. 61–65/ 8, 10, 12, 13, 14, 23. These do not need to be done prior to the Oct. 18 midterm. Of these, hand in only #s 8, 12, 13, and 23. Prior to doing #13, do non-book problem B4. In #s 12 and 13, the Proposition on p. 48 ("Sequences in \({\bf R}\) `behave well' with respect to arithmetic") is helpful.
Definition for #8: A reordering of a sequence \( (p_n)_{n=1}^\infty \) is a sequence \( (q_n)_{n=1}^\infty \) such that for some bijection \( f:{\bf N}\to {\bf N} \) we have \( q_n=p_{f(n)} \) for each \(n\in{\bf N}\). Problem 8 is asking you to prove that if \( (q_n)_{n=1}^\infty\) is a reordering of a convergent sequence \( (p_n)_{n=1}^\infty\) in a metric space, then \( (q_n)_{n=1}^\infty\) converges and \(\lim_{n\to\infty} q_n = \lim_{n\to\infty} p_n\).- B: Click here for non-book problems (updated 10/21/16). Of these, hand in only B4, B5(b), and B7. (For purposes of writing up the last part of B7, you may assume the result of B6.) To be prepared for the first exam, you should have B1–B3 done by Friday 10/14/16.
General homework page
Class home page