Aug. 31 | Sept. 7 | Sept. 12 | Sept. 19 | Sept 26 | Oct 3 | Oct 10 | Oct 17 | Oct 24 | Oct 31 | Nov 7 | Nov 14 | Nov 21 | Nov 28 | Dec 5

Final Exam

- Aug. 30: (some of this is a bit ahead of us, but we'll get to it next time)

- Chapter 1.1 #2abc, 3ab, 5a, 6b, 9
- Chapter 1.2 #1,2,3a,4a,6c,7a
- Suppose
*f*is a function. Is*f*(*A*_{1}*A*_{2}) =*f*(*A*_{1})*f*(*A*_{2}) for all sets*A*_{1}and*A*_{2}that are subsets of the domain of*f*? If true, prove it. If false, find a closely related result that is true, and prove that.

- Sept. 1:
- [You should be reading the text before the problems...]
- Chapter 1.2 #3b,4bc,6a, 10"a", 11 (we did the second part of 10 in class)

- Sept. 6:
- Read chapter 1.4, do 1.4 #1a,2ab,3ab,5abcgi, 8,10,12,13
**For Thursday, turn in 1.1#5a,***f*(*A*_{1}*A*_{2})*f*(*A*_{1})*f*(*A*_{2})*A*_{1},*A*_{2}subsets of the domain of*f*, 1.2 #3a,11

- Sept. 8: Chapter 1.4 #14bc,15bc,21, read 1.5, do #1,3,4,6a

- Sept. 13: Chapter 1.5 #5,7,8, 1.7 #1a (optional for fun: 1.6 #1a,2a),

- Sept. 15: Chapter 1.7 #3,4a (these are
*intervals*, not ordered pairs), 6a,7,8ab,13,14

- Sept. 20: Do 1.7 #8ef,9,11,12,16,17 and enjoy the story: Welcome to the Hotel Infinity!
- Sept. 22: Read 2.1,

- do 2.1 #2 (the "backwards triangle inequality" - hint: use the regular triangle inequality to prove this),

- 2.1 #6a,b,d (hint: interpret each absolute value as a distance, in words - then the result is easier to see),
- Do Metric Space handout section 1, problems #1-3
- Read 2.2

- Sept. 27:
**Quiz 1**; Do 2.1 #3,8ac,9a,10,11a,15 and 2.2 #1,2,3 - Sept. 29: Do 2.2 #6acd,7a,c,9,

- Oct. 4:
**For Thursday, turn in 1.4#3b,5a,8,14b,15b 1.7#7, 2.1#2,8a, 2.2#1** - Also prove: If
*A*has supremum , then there is an increasing sequence {*a*} of elements of_{n}*A*that converges to .

[It may be helpful to dispose of the case that*A*first, then do the other case; otherwise the "increasing" can be confusing.]

Hint:*n*, - 1/*n*is*not*an upper bound of*A*(why?)

- Oct. 6:

- Oct. 11:
**Exam 1** - Oct. 13: Do Ch. 2.4 #1,2,3ace,4a,7abde,11. Read 2.5 and section 2 of the metric space supplement

- Oct. 18:
- Oct. 20:

- Oct. 25:
- Oct. 27: Do 3.1 #4,5,8,11,12,14,

- and try to "prove" int(
*A**B*) = int(*A*)int(*B*) (which is in fact false) to see how the argument, which worked so well for int(*A**B*) = int(*A*)int(*B*), breaks down in the union case.

- Nov. 1:
- Nov. 3:

- Nov. 8:
- Nov. 10:

- Nov. 15:
- Nov. 17:

- Nov. 22:
- Nov. 24:
**Thanksgiving Break, no classes**

- Nov. 29: Do 4.1 #1a,2a,6,7a, 4.2 #6a,11,21, and

- Give an epsilon-delta proof that if
*f*and*g*are functions from to which are continuous at*x*=*a*, then*f*+*g*is continuous at*x*=*a*, and - Use the inverse-image / open-sets characterization of continuity to prove that if
*X*,*Y*,*Z*are metric spaces with*f*:*X**Y*and*g*:*Y**Z*being continuous functions, then*g*o*f*is continuous also. Hint: Recall that (*g*o*f*)^{-1}=*f*^{--1}o*g*^{-1}.

- Dec. 1:

- Dec. 6:
- Dec. 8:

- Friday, Dec. 16, 2-5 p.m..
**Final Exam**

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