Due date: Friday, Oct. 31
Last update made by D. Groisser Mon Oct 27 13:20 EDT 2014
You are required to do all of the problems and reading below. You will not be required to hand in all the problems. I’ve indicated below which problems 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.
- C: (1) Re-take the proof-writing quiz on the miscellaneous handouts page as many times as you need in order to get 100% consistently. (2) Re-read the handout “Mathematical grammar and correct use of terminology”. Pay special attention to the discussion of the phrase “such that” on p. 3 (item 5). (3) Go through all the comments on your exam and make sure you understand every one. If there is a comment you don’t understand, see me in office hours as soon as possible.
- A: Rosenlicht pp. 61–65/ 8,10,12,13,14,23. Prior to doing #13, do non-book problem B7. Of these, hand in only #8.
Also, there will be a step in the proof for “an + bn” that involves the idea “Let N = max{N1, N2}.” For this proof, write out this step in detail. After this homework, assuming you all do this part of the problem essentially correctly, I will allow you to abbreviate this type of argument the way discussed in class.Definition for #8: A reordering of a sequence {pn}n=1∞ is a sequence {qn}n=1∞ such that for some bijection f: N→ N we have qn = pf(n) for all n∈N. (In “{pn}n=1∞“, the superscript “∞” is meant to be sitting over the subscript “n=1“, but I don’t know how to achieve this in HTML.) Problem 8 is asking you to prove that if {qn}n=1∞ is a reordering of a convergent sequence {pn}n=1∞ in a metric space, then {qn}n=1∞ converges and limn →∞ qn = limn →∞ pn.
- B: Click here for non-book problems (updated 10/23/14). Of these, hand in only B1a, B2, B4a, B4b(i)(ii), B6, B7, and B9. In B7, there will be a step in the “(ii) implies (i)” proof that involves the idea “Let N = max{N1, N2}.” For this proof, write out this step in detail. After this homework, assuming you all do this part of this problem essentially correctly, I will allow you to abbreviate this type of argument the way discussed in class.
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