Last updated Sun Oct 9 20:42 EDT 2016
Due-date: TBA
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:
- Finish reading all parts of Rosenlicht Chapter II of Rosenlicht that we did not cover in class.
- Read the handout "Interiors, Closures, and Boundaries" posted on the Miscellaneous Handouts page. (You may ignore fact #13 until we've defined convergent sequences.) Several of the facts on this handout were proven in class. Proofs of the others will be assigned as part of the next homework. All of these facts (except #13) and their proofs will be fair game for the first exam.
- A: Rosenlicht pp. 61–63/ 1c, 3, 4, 6, 7. Of these, hand in only 3, 4, 6.
Note:
- In problems like 4 and 6, keep in mind that "proof by picture" is not a valid method of proof. In these two problems, you will need to show algebraically that open balls of certain centers and radii (which you have to figure out) are contained in certain sets. In these problems you will be tempted to use the concept of "the point (or a point) on a graph that's closest to a given point not on the graph." But you can't assume there is a closest point, unless you have a written proof that such a point exists. (For #6, it's highly unlikely that any attempts you make to prove the existence of such a point will succeed; you don't have the tools yet. Once you do have the tools, later in this course, you'll see that it's circular reasoning to try to use the closest-point idea to prove that the set in this problem is open.)
- In Rosenlicht, En means Euclidean n space: the metric space (Rn, d), where d is the Euclidean metric. (See the last paragraph on p. 34.) So in problems 4 and 6, E2 is the usual xy plane (or x1x2 plane) with the distance-formula that you're used to. In these problems, you may use the notation (x, y) instead of (x1,x2), but state that you're doing this (if I have you hand in one or both of these problems), so that I know what you mean from the start.
- B: Click here for non-book problems (updated 10/4/16). Note that B1 is an expanded version of Rosenlicht's problem 1b. Of these, hand in only B1cd (you may assume the results of parts (a) and (b)), B3, B4, B5. Note: Only B1–B5 are part of the non-book problems for this assignment. B6–B8, which I had temporarily added to this assignment, have been moved to the next assignment. I thought I had re-posted the correct (B1–B5) list of problems 20 minutes after I posted the B1–B8 list, but apparently I left the longer list on the website. I apologize for any confusion.
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