Last updated Mon Nov 15 00:41 EST 2021
Due-date: Friday, 11/19/21
You are required to do all of the problems and reading below (except for anything explicitly labeled "optional"). You will not be required to hand them all in. I have indicated below 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 are to 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: Abbott exercises. See notes and additional instructions below this list before starting these problems.
- 2.6/ 2 (posted 11/4/21)
- 2.7/ 2, 5, 9, all posted 11/5/21. The Section 2.7 exercises I posted on 11/4/21 were from an earlier edition of the textbook, and several of them don't exist in the current edition. (I was anxious to get some exercises posted for you, but didn't have my physical copy of the book with me, and used a previously-downloaded electronic copy that I didn't realize wasn't the current edition.)
- 3.2/ 1, 2 (both posted 11/5/21)
- 3.2/ 8, 10 (both posted 11/13/21)
- 3.3/ 4, 5, 8 (all posted 11/14/21)
Of the Abbott problems, hand in only these:
(See end of non-book problem B6 to make sure you understand what needs to be proved.)
- 2.7/ 2e, 5
- 3.2/ 2a, just for the set \(A\). Prove your answer.
- 3.3/ 4a (prove your answer)
Notes and additional instructions on some of these problems:
- 2.7.2. Review Corollary 2.4.7 first. (This was assigned reading in the previous assignment.) You will prove this corollary in exercise 2.7.5, but may use it for the sake of producing examples in these earlier exercises; the reasoning is not circular.
- 2.7.9. The convergence-test that is usually called the "Ratio Test" says more than what is stated in this exercise. See non-book problem B4.
- All exercises from sections 3.2 and 3.3: at the start of each set of exercises, Abbott should have said: "In these exercises, `set' means `subset of \(\bf R\)' unless otherwise specified.
- 3.2.2. Read Definition 3.2.6 before doing part (c).
- 3.2.8. (i) "Determine" here means that you should be able to prove your answer. (ii) "Neither" here means "not definitely closed and not definitely open;" i.e., that there is neither enough information to determine a definite answer to the question "Is this set closed?", nor enough enough information to determine a definite answer to the question and "Is this set open?" (iii) For part (e), a useful observation is that for any set \(S\subseteq {\bf R}\), we have \(S\subseteq \overline{S}\), which also implies that \(S^c\supseteq (\overline{S})^c\).
- 3.3.4. (i) The meanings of "Determine" and "neither" are analogous to their meanings in Exercise 3.2.8.
- 3.3.8(b). A set \(D\subseteq{\bf R}\) is called discrete if every element of \(D\) is isolated point of \(D\). Suggestion for doing Exercise 3.3.8(b): consider unbounded discrete sets \(K\) and \(L\).
The definition of "distance between two nonempty subsets of \(\bf R\)" given in this exercise applies whether or not the sets are compact. But the distance between two sets can be zero without the sets being disjoint, as part (b) shows. If we don't impose the condition that both sets be closed, it's much easier to find an example of a pair of disjoint sets \(A,B\) for which the distance between \(A\) and \(B\) is \(0\). For example, we can take \(A=(-\infty,0)\) and \(B=[0,\infty)\).
B: Click for non-book problems. Posting-date(s), with updated numbering of problems: 11/5/21 for B1, B2, and B4; 11/6/21 for B5; 11/8/21 for B3; 11/13/21 for B6.
Of these, hand in only B6.
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