Due date: Friday, Oct.9 (but everything on this assignment is fair game for the Wednesday, Oct. 7 exam).
Last update made by D. Groisser Wed Oct 7 20:35 EDT 2015
You are required to do all of the problems below. You will not be required to hand them all in. I’ve indicated below which ones you do have to hand in (fewer than usual to give you a little break after the exam). 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/ 1bc, 3, 4, 6, 7. Of these, hand in only 3, 6. Note:
- In 1b: a sequence (xn) of real numbers is bounded if there exists M∈R such that |xn| ≤ M for all n∈N.
- 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. For the notation in B6(b) and B8, see the “Interiors, Closures, and Boundaries” handout. Of these, hand in only B6a.
- C: 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. Prove all the others for homework. Of these, hand in only the proofs of facts 10 and 21.
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