Last updated Tue Oct 16 13:31 EDT 2018
"Virtual" due-date: Friday, 10/12/18. No homework will be collected for this assignment. (This updates the previous notice that some hand-in problems might be required for this assignment, just at a later date. I would rather have you move on to the next homework assignment.)
You are required to do all of the problems below. You will not be required to hand them all in. I will announce later 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" (henceforth "the ICB handout") posted on the Miscellaneous Handouts page. (You may ignore fact #13 until we've defined convergent sequences. Fact #13 isn't fair-game material for your first exam.) Prove all the facts, other than #13, that have not been proven in class as of Wed. 10/10/18. The facts not proven in class will include numbers 5, 10, and several from the 14–22 group. (By Wed. 10/10, facts 11 and 12 will have been proven in class.) On your Oct. 15 exam:
- You will be expected to know, and be able to use, all the definitions in the ICB handout (all of which occur in the "Definitions" paragraph near the beginning).
- You will be expected to know facts 1–10 (of which #6 is just a repeat of a definition).
- You will not be expected to know the remaining facts, but you'll be expected to be able to prove any fact in the ICB handout (henceforth "ICB fact"), since the proofs are fairly simple applications of definitions. However, on the exam, using ICB facts to prove other ICB facts is subject to some restrictions:
- For facts other than facts 1–10, you will be allowed to use facts 1–10.
- If asked to prove any ICB fact that was proven in class (this includes most of 1–10, and by Wed. 10/10 will include several more), you will be expected to prove it again, using only the relevant definitions (and facts we proved before making these definitions), not by using other ICB facts. When preparing for the Oct. 15 exam, I suggest that you not waste time memorizing all the ICB facts after #10, let alone which of these were proven in class. During the exam, you may ask me (quietly) "Am I allowed to use such-and-such fact?" (Note: If "such-and-such fact" is false, I'm not going to tell you; I'll only say that you're not allowed to use it.)
- 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.) You are likely to find #6 difficult.
- 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 B2 is an expanded version of Rosenlicht's problem 1b.
In class I gave a hint for B6. The same hint is helpful in B8. Also helpful are a couple facts on the "Interiors, Closures, and Boundaries" handout that will be proven on Wed. 10/10/18.
General homework page
Class home page