Due date: Wednesday, 10/15/14
Last update made by D. Groisser Sat Oct 11 01:27 EDT 2014
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 such a point exists, 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 (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 (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. Of these, hand in only B3.
- 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 on Friday Oct. 3; some others will be proven on Mon. Oct. 6. Prove all the others for homework. I recommend that you not wait till Monday to see which of the “Boundaries” facts I prove in class before you start working on these. All of this material will be fair game for the Friday Oct. 10 exam, so proving these facts on your own could be useful preparation. You do not have to hand in any of these.