Last updated Fri Oct 26 00:01 EDT 2018
Due-date: Wednesday, 10/31/18.
"A ghoul ate my homework" will not be accepted as an excuse.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.
A: Rosenlicht Chap. III/ 8, 10, 12, 13, 14, 23, 24. See "Notes" below before starting these problems. Of these, hand in only #s 8, 12, 13, and the last part of 23. Notes:
- 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 \(\lim_{n\to\infty} q_n = \lim_{n\to\infty} p_n\).
- Prior to doing #13, do non-book problem B1.
- In #12 and #13, the proposition on p. 48 ("Sequences in \({\bf R}\) `behave well' with respect to arithmetic") is helpful.
- In #14, Rosenlicht's hint is very useful, but you still have to write out the argument for the case that he says is "easy"; you can't simply jump to the assumption in the second half of the hint's sentence.
B: Click here for non-book problems. Of these, hand in only B1, B2(b) parts (i) and (iv), B4, and B5. For purposes of writing up B4, you may assume the result of B3.
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