Due date: Friday 11/13/15. For this assignment, the “due date” 11/13/15 is the date by which you should finish doing all the problems. The hand-in date will be Monday 11/24/15. The hand-in problems are announced below.
Last update made by D. Groisser Thurs Nov 19 17:26 EST 2015
You are required to do all of the problems and reading below. You will not be required to hand in all the problems. I’ve indicated below which problems you do have to hand in. 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. 61–63/ 24, 26 (prove your answer), 28, 29, 31, 32. (“31” was mistyped as “33” in first posting of this assignment, here and in the note below.) Suggestion for simplifying #32: prove that the union of two compact subsets is compact, and use induction. Of these, hand in only 26, 28, and 32.
- B: Click here for non-book problems. Of these, hand in only B2.
Note: #31 gives an efficient proof that a closed interval [a,b] in R is compact. In stating the problem, Rosenlicht only outlines this proof. In the next-to-last sentence, he doesn’t say why 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 Ui is open to help you show that l.u.b.(S)= b.”
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