Last updated Fri Oct 27 12:24 EDT 2017
Due-date: Friday 11/3/17
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. Of these, hand in only #s 8, 12, 13, 14, and 23. Prior to doing #13, do non-book problem B1. In #s 12 and 13, the Proposition on p. 48 ("Sequences in \({\bf R}\) `behave well' with respect to arithmetic") is helpful. In #14, I strongly recommend using Rosenlicht's hint, 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.
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. Of these, hand in only B1, B3(b), and B5. (For purposes of writing up B5, you may assume the result of B4.)
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