MAA 4211 Assignment 4
Due date: Monday, 10/7/13
Last update made by D. Groisser Fri Oct 4 15:25:28 EDT 2013
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. 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 1c, 3, 4, 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 Rosenlicht, En means Euclidean n space: the metric space (Rn, d), where d is the Euclidean metric (the square root of the sum of the squares of differences of coordinates). (See the last paragraph of 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, so that I know what you mean from the start.
- B1: (Hand this one in.) Define a metric d on the set of rational numbers Q by d(x,y) = |x – y| (the restriction to Q of the standard metric on R). Give an example, with proof, of a nonempty, proper subset of (Q,d) that is both open and closed in this metric space. (Do not expect your subset to be either open or closed in R, let alone both open and closed. There is no nonempty, proper subset of R that is both open and closed with respect to the standard metric.)
- 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.) Prove facts (4)-(5), (7)-(10), and (14)-(21) stated in the handout. Of these, hand in the proofs of only (9), (10), (16), (20), and (21). When working on any of these, you may assume any of the facts listed earlier in the handout, but not those listed later. On Friday Oct. 4, I’ll prove facts (11) and (12), and possibly some of the others from (7)-(21), but don’t wait to see which ones I do in class to try doing these on your own.