Last updated Thu Oct 12 22:38 EDT 2017
Due-date: Wednesday, 10/18/17 (the day of the first midterm)
You are required to do all of the problems below. To allow you time to study for the midterm, I am not asking you to hand in any problems on this assignment.
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.) Prove all the facts, other than #13, that were not proven in class. All facts on this handout (except #13), and their proofs, will be fair game for the first exam.
- A: Rosenlicht Chap. III/ 1c, 3, 4, 6, 7.
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, \(E^n\) means Euclidean \(n\)-space: the metric space \( ({\bf R}^n,d)\) where \(d=d_{\rm Euc}\) is the Euclidean metric (the one induced by the \(\ell^2\)-norm). (See the last paragraph on p. 34.) So in problems 4 and 6, \(E^2\) is the usual \(xy\)-plane (or \(x_1x_2\)-plane) with the distance-formula that you're used to. In these problems, you may use the notation \((x,y)\) instead of \((x_1,x_2)\), 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. Problem B1 is an expanded version of Rosenlicht's problem 1b. Contrary to my original instruction, you do not need to wait until we've defined sequences in class to get started on parts (b)–(e) (the problem defines everything you need).
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