Math 301: Real Analysis

Spring 2003

Schedule

Homework is due once a week, unless stated otherwise.

Monday, February 3, 2003

• Homework: Read sections 1-4 in Ross. Do problems: 1.4, 1.7, 1.8, 1.9, 1.11; 2.3, 2.6; 3.1, 3.4.

Thursday, Feb. 6, 2003 Book Work Quiz on the definition of an algebraic number and the statement and proof of the rational zeros theorem.

Monday, February 10, 2003

• Homework: Read sections 4-7 in Ross. Do problems: 3.5, 3.6, 4.1(a-j), 4.2(a-j), 4.3(a-j), 4.4(a-j), 4.6, 4.14

Thursday, Feb. 13, 2003 Book Work Quiz with five questions: (1) Prove Thm 3.1 (ii) and (iii). (2) Prove Thm 3.2 (i) and (ii). (3) For a non-empty set S in the real numbers, give a definition of an upper bound of S, a lower bound of S, the sup of S, and the inf of S. (4) State the completeness axiom. (5) I will ask you to find examples of sets S in the real numbers that have certain properties with respect to their upper bounds, lower bounds, sup, and inf. For example, I might ask you to find a set where the sup is a real number in the set and the inf does not exist as a real number.

Monday, February 17, 2003

• Homework: Read sections 7-10 in Ross. Do problems: 7.2, 7.3, 7.4, 7.5, 8.2, 8.4, 8.6, 9.1

Thursday, Feb. 20, 2003 Book Work Quiz with 2 questions: The first question will ask you to define the limit of a sequence and the second question will ask you to state and prove one of the following three theorems: Example 6 p. 41, Theorem 9.1, or Theorem 9.4. You must know the statements and proofs of all three.

Monday, February 24, 2003

• Homework: Read sections 10-12 in Ross. Do problems: 9.9, 9.10, 9.17, 9.18, 10.1, 10.2, 10.10, 11.1,11.2, 11.4 11.10

Monday, March 3, 2003

• Homework: Read sections 10-13 in Ross. Do problems: 10.6, 10.8, on the Handout do problems 6.3 and 6.4. Then in the book again do problems 12.3, 12.4, 13.2, 13.4, 13.6, 13.9,13.10.

There is no QUIZ this week but make sure you know well and understand the definitions of limsup s_n and liminf s_n as well as the definition of a Cauchy sequence. Also go over the main theorems about these concepts.

The First Exam, a take-home, is coming up during the weekend of March 7-10.

Friday, March 14, 2003

• Homework: Read sections 14-16 in Ross. Do problems: 14.1, 14.2, 14.5, 14.6, 14.12, 14.13, 15.1, 15.2, 15.4, 15.6, 16.1, 16.3, 16.4

Monday, March 31, 2003

• Homework: Read sections 17,18,19 in Ross. Do problems: 17.1,17.5,17.6,17.9,17.10,17.13, 18.2, 18.4

Monday, April 7, 2003

• Homework: Read sections 18,19,20,23 in Ross. Do problems: 18.6 (you may assume that cos(x) is continuous),18.9,18.12,19.1, 19.2, 19.4, 19.5, 20.1, 20.5

QUIZ ON THURSDAY April 3

Monday, April 14, 2003

• Homework: Read sections 23 and 24 in Ross. Do problems: 23.1, 23.3, 23.4, 23.7, 23.8, 23.9, 24.1, 24.2, 24.6, 24.10, 24.11, 24.13

QUIZ ON THURSDAY APRIL 10

Monday, April 21, 2003

• Homework: Read sections 25 and 26 in Ross. Do problems: 25.6, 25.8, 25.9, 25.12, 26.6, 26.7, 26.8

QUIZ ON THURSDAY APRIL 17: State and prove Thm 24.3. Three questions about continuity, uniform continuity, pointwise convergence, and uniform convergence. So know all the definitions and go over homework on those questions.