Last updated Mon Nov 7 19:51 EST 2016
Due-date: Monday, Nov. 14
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 pp. 61–65/ 24, 26 (prove your answer), 28, 31, 32, 33. Suggestion for simplifying #32: prove that the union of two compact subsets is compact, and use induction. In #33, replace the term "cluster point" with "accumulation point". Of these, hand in only 26, 28, 32, and 33.
Note: #31 gives an efficient proof that a closed bounded interval \([a,b] \subset {\bf R}\) is compact. In stating the problem, Rosenlicht only outlines this proof. In the next-to-last sentence, he doesn't say why \({\rm l.u.b.}(S)\) exists; that's one of the details you have to fill in. In the last sentence, there are several details to fill in. If this were the last sentence of a proof handed in by a student, I'd give very little credit, and would write a comment like, "How does this follow from anything you've written or that we've proved?" or "Practically the whole proof is consists of justifying this statement!" Don't expect to understand the last sentence of the problem when you read it. That sentence is essentially a hint that could be worded as, "Use the fact that \(U_i\) is open to help you show that \({\rm l.u.b.}(S)=b.\)"
- B: Click here for non-book problems (updated 11/7/16). Of these, hand in only B1, B4, B7abcde.
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