Due date: Wednesday, 9/23/15
Last update made by D. Groisser Sat Sep 19 01:50 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.
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: Finish reading all of Chapter II of Rosenlicht, since we’re skipping over a lot in class. (You do not need to finish this before starting the problems in part C.)
- B: Read the handout “One-to-one and onto: What you are really doing when you solve equations” posted on the Miscellaneous Handouts page. The logic of the first two pages (up to but not including the paragraph, “What this has to do with `one-to-one’ and `onto’ “) applies also to solving inequalities, such as the ones in Rosenlicht problem #5 below.
- C: Rosenlicht pp. 29–31/ 4a (figure out a way to do this that does not require any division computations, works for any ordered field, and does not use a calculator), 5-11, 13, 14. Of the problems above, hand in only 5c, 6, 10abc, 11, 13, 14.
Comments on some of these problems:- I suggest doing #11 before #10. You may find the result of #11 useful in proving the answers to one or more parts of #10.
- In #11, I strongly suggest ignoring Rosenlicht’s hint. There is a much faster way to proceed.
- In #10, replace the instruction, “giving reasons if you can” with “prove your answer.”
If you find part 10(c) much harder than (a) or (b) (especially the “prove your answer” part), you are not going crazy! In fact, if you do not find it hard to prove your answer, you are probably implicitly assuming some fact we haven’t proved. If you think you have a proof, keep in mind that we have not defined what a limit is, let alone proved any properties about limits. All we have is the Least Upper Bound property of R. You need to find a way to prove your answer that does not implicitly or explicitly assume something about limits. (That doesn’t mean that you can’t use ideas about limits to help you guess the l.u.b. But proving that your guess is correct can’t use the word “limit”, at this stage of the course. We’re still in Chapter II; limits are a topic in Chapter IV. Once we cover limits, you’ll be allowed to use the theorems we prove..