I am usually available MWF after class, and sometimes 12:45-1:45 or around 11:15 am; but I need to know in advance if you'd like to meet with me. Professor Tollisen is available MTThF, 2-5pm in Library 17.
Final Exam: F 12/8, 1:00 pm. The exam will cover the whole semester, with some emphasis on the material covered after Midterm 2.
HW 26, due F 12/1. Do Sec 2.3: 12; HW7: X4; HW 8: X6; HW 14 X1; Prove the alternating series test.
HW 22, due M 11/13. Prove Theorem 5.21 using the definition of compactness (every sequence has a subsequence that converges to a point in the set); also do Sec 5.3: 10, 12, 14.
HW 21, due F 11/10. Read Sec 5.3; may skip proofs of Theorems 5.21, 5.27, 5.30. Prove Theorems 5.22 and 5.31, and Examples 5.26 and 5.32, on your own. Also do Sec 5.3: 1-3, 5, 6, 9, and X1: Prove f(x)=2x is uniformly continuous on R (without resorting to Example 5.25).
HW 20, due W 11/8. Read Sec 5.2; may skip p.109 (we may come back to it later, though). Do Sec 5.2: 1-4, 8; also prove Proposition 5.14 on your own; and do: X1.Does there exist a set of open intervals such that the sum of their lengths is finite, and their union contains all rational numbers? For #2, first give a definition of f((a,b)), and for your answer just explain your reasoning without a rigorous proof; for #4, first give a definition for "extend"; for #8, just give counterexamples, without proof.
HW 19, due M 11/6. Read Sec 5.1. Do Sec 5.1: 1, 2, 4, 6, 8, 9; no proof necessary for 8 & 9.
HW 18, due M 10/30 F 11/3. Read Sec 4.5. (In Definition 4.43, "it is possible to find" only means "there exists"; it doesn't mean we must be able to find two such open sets.) Do Sec 4.5: 3, 4, 5, 7; prove the "only if direction" of Theorem 4.46 without looking at the book's proof. Also do: X1. Is Q connected, disconnected, or totally disconnected? Prove your answer. Hint for 3: First do X1. For #5, explain your reasoning, but a rigorous proof is not necessary.
Midterm 2, W 11/1. The exam will cover HWs 9 through 18 17.
HW 17, due W 10/25 F 10/27. Read Sec 4.4; may skip proof of Theorem 4.42. Do Sec 4.4: 1, 3, 5, 6; X1: Prove all three assertions in the last paragraph of Sec 4.4: (a) Every open set in R is the countable union of open intervals; (b) There is a closed set that is not the disjoint union of closed intervals; (c) Every closed set is the countable intersection of sets each of which is the union two disjoint closed intervals.
HW 16, due F 10/20 M 10/23. Read Sec 4.3; may skip proofs of Theorems 4.32 and 4.37. Do Sec 4.3: 1, 3, 4, 6, 8, 9; X1: Prove between any two irrational numbers there is a rational number. For #3, first give a definition for the distance between two disjoint subsets of R; also, show that the statement is not necessarily true if K is closed but not compact. For #6, the hypothesis that |aj - bj| -> 0 is not necessary (but maybe it makes the problem easier? I don't see how, though). In #8, cK means R \ K. In #9, the statement is different than what Proposition 4.39 says; do you see the difference? Hints for #6 and #9.
HW 15, due W 10/18. Read Sec 4.2. Do Sec 4.2: 1-7. Also prove Propositions 4.21, 4.24, 4.25 on your own. I suggest you first do problems 2-5, 7, then 1, 6. For #6, first prove: x is an accumulation point of S iff every neighborhood of x contains a point in S distinct from x.
HW 14, due M 10/16. Prove Propositions 4.5, 4.6, and 4.12: first try to prove them before reading the proofs in the book; if you don't succeed, then read the proof in the book and later (after a few hours or the next day) prove it yourself. Also do Sec 4.1: 8, 9; and X1: Prove, without using Proposition 4.13, that a subset S of R is closed iff every convergent sequence in S converges to a point in S. Hint: you might get some ideas from the proof of Prop 4.13. Prove your answers for both #8 and #9.
HW 13, due F 10/13. Read Sec 4.1; may skip proof of Proposition 4.7 and 4.13. Do Sec 4.1: 1-3, 5-7.
HW 12, due W 10/11. Read Sec 3.4 p.63-65; may skip proofs of Theorems 3.44 and 3.46. Do Sec 3.4: 2, 3, 7; Sec 3.3: 10; also do these extra problems. For #7, use #6 (even though we are not doing #6 --- it's too hard).
HW 11, due W 10/4 F 10/6. Read Sec 3.3; may skip proofs of all the theorems. Do Sec 3.3: 1, 2, 8. (Even though 8 is starred, I think it's neither very difficult nor very long.) Also do: X1: Prove that if an infinite series converges conditionally, then the sum of its positive terms diverges.
HW 10, due M 10/2. Read Sec 3.2; may skip proofs of Theorems 3.13 and 3.20. Prove Theorem 3.15 (Geometric Series) without looking at the book's proof. Give a detailed proof of Theorem 3.19 (Root Test), including justification for the second sentence in the book's proof. Also do Sec 3.2: 4-10.
HW 9, due F 9/22 M 9/25 F 9/29. Read Sec 3.2 p. 47. Do Sec 3.1 (not 3.2): 2, 6, 7; may use Comparison Test for them. Also prove Proposition 3.10 without looking at the book's proof.
Midterm 1, W 9/27. The exam will cover HWs 1-8.
HW 8, due W 9/20. Read Sec 3.1. Do Sec 3.1: 3-5; also do these extra problems.
HW 7, due M 9/18. Read Sec 2.4. Do Sec 2.4: 2, 3. Also do these extra problems.
HW 6, due F 9/15. Read Sec 2.3; may skip proof of Theorem 2.36. Do Sec 2.3: 1, 4, 5, 6, 7, 10, 12. Also do these extra problems. For #7, prove your answer only for lim sup (aj + bj); for the other parts of the problem, just answer without proof. For #10, first give a precise definition of what "arbitrarily large" means in Proposition 2.39.
HW 5, due W 9/13. Read Sec 2.2; may skip proof of Theorem 2.24. Do Sec 2.2: 1, 2, 3, 5, 6. Also do Sec 2.1: 11. If you're stuck on a problem after multiple tries, here are some hints.
HW 4, due M 9/11. Read Sec 2.1 p.22-27; may skip the second part of the proof of Theorem 2.12 (where it says "Conversely"). Do these problems.
HW 3, due F 9/8. Read Sec 2.1 p.19-22. Do Sec 2.1: 1, 7, 8, 10, 13. For #8, prove only part (2), not (4), of Proposition 2.6.
HW 2, due W 9/6. Read Sec 1.2. Do Sec 1.2: 1-6. For #4, prove "commutative" only. For #6, see the definition and theorem on p.329 in the Appendix.
HW 1, due F 9/1. Read Sec 1.1 p.1-5. Do Sec 1.1: 1-5, 7, 8, 11. For #11, describe an infinite countable set of irrational numbers between 0 and 1. Also do: X1: Prove, without using Theorem 1.8, that there exists a real number c such that c^2 = 2.