MATH 621 Homework

Homework Assignments

12/05
  1. Read the section “Integration and Differentiation” in Rudin.
  2. Complete Exercises 9, 10(d), 13 from Chapter 6 of Rudin.
  3. Update (12/09) Your final exam consists of 8 exercises from Rudin: C1-17, C2-19, C2-29, C3-14, C4-7, C5-22, C6-6, C6-17. (Notation: C1-17 refers to Exercise 17 from Chapter 1.) All but one of these have been assigned at the appropriate point in the semester, so you may have solutions to them already written. If so, please take the time to look over your solution(s) in light of your later experience in the class, and re-write them as you see fit. The exam is due at the end of the scheduled final exam time for the class: Friday, December 22, at 12:00 noon.
12/05
  1. Read the proof of Lebesgue’s Theorem from Conway’s “A First Course in Analysis” (handed out in class), and finish reading the section “Properties of the Integral” in Rudin.
  2. Complete Exercises 1, 2, 6, 7, 8 from Chapter 6 of Rudin.
11/21
  1. Have a good Thanksgiving!
  2. Finish reading Chapter 5 of Rudin.
  3. Complete Exercises 25, 26, 27, 28 from Chapter 5 of Rudin.
  4. Hand in Exercises 25, 27 from Chapter 5. Due 12/05.
11/16
  1. Read the next 3 sections of Chapter 5 (Mean Value Theorems, Continuity of Derivatives, and l’Hospital’s Rule). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 1, 7, 8, 9, 22, 23 from Chapter 5 of Rudin.
11/14
  1. Finish reading Chapter 4 in Rudin, and read the first section of Chapter 5 (The Derivative of a Real Function). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 13, 20, 21, 22, 23 from Chapter 4 of Rudin.
  3. Hand in Exercises 22, 23 from Chapter 4. Due 11/21.
11/07
  1. Read the next two sections of Chapter 4 in Rudin (Discontinuities, and Monotonic Functions). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 5, 7, 9, 11 from Chapter 4 of Rudin.
  3. Hand in Exercises 9, 11 from Chapter 4. Due 11/14.
10/31
  1. Read the next two sections from Chapter 4 in Rudin (Continuity and Compactness, and Continuity and Connectedness).
10/26
  1. Read the next section from Chapter 4 in Rudin (Continuous Functions).
  2. Begin working on the midterm exam, which is due 11/07.
10/24
  1. Read the first section from Chapter 4 in Rudin (Limits of Functions).
10/19
  1. Finish reading Chapter 3 of Rudin. Come to class on Monday armed with any final questions from either this reading or anything we have covered so far in the course.
10/17
  1. Read the next two sections of Chapter 3 in Rudin (The Root and Ratio Tests, and Power Series), and also the sections on Absolute Convergence and Rearrangements. (These last two sections cover material we went over in class last week.) You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 23, 24, 25 from Chapter 3 of Rudin.
  3. Hand in Exercise 24 from Chapter 3. Due 10/24.
10/10
  1. Read the next two sections of Chapter 3 in Rudin (Series of Non-negative Terms, and The Number e). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 11, 12, 14, 16 from Chapter 3 of Rudin.
  3. Hand in Exercises 7, 12, 16 from Chapter 3. Due 10/17.
10/05
  1. Read the next three sections of Chapter 3 in Rudin (Upper and Lower Limits, Some Special Sequences, and Series). Send me a question on the reading by Monday evening. Due 10/09.
  2. Complete Exercises 4, 5, 6, 7 from Chapter 3 of Rudin.
10/03
  1. Read the next two sections of Chapter 3 in Rudin (Subsequences, and Cauchy Sequences). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Complete Exercises 1, 2, 3 from Chapter 3 of Rudin.
  3. Hand in Exercises 11, 30 from Chapter 2; Exercise 3 from Chapter 3. Due 10/10.
09/28
  1. Read the first section of Chapter 3 in Rudin (Convergent Sequences). You should find this section a relief after Chapter 2, so you do not need to send me a question but, as always, feel free to do so if anything is not clear.
  2. Complete Exercises 11, 29, and 30 from Chapter 2 of Rudin. (You could also look at Exercises 27 and 28, the first of which perhaps explains the importance Rudin attaches to perfect sets, but you do not need to complete them.)
09/26
  1. Read the last two sections of Chapter 2 in Rudin (Perfect Sets, and Connected Sets). You do not have to send me a question on this reading, but feel free to do so if anything is not clear.
  2. Complete Exercises 17, 19, 20, 22, 23, 25 from Chapter 2 of Rudin.
  3. Hand in Exercises 12, 14, 22, 25 from Chapter 2. Due 10/03.
09/21
  1. Read the next section of Chapter 2 in Rudin (Compact Sets). Send me a question on the reading by Monday evening. Due 09/25.
  2. Complete Exercises 12 through 16 from Chapter 2 of Rudin.
09/19
  1. Read the next section of Chapter 2 in Rudin (Metric Spaces). You do not have to send me a question on this reading, but feel free to do so if there is anything you feel needs clarification, or that you would like to discuss in class.
  2. Work on Exercises 5 and 6 of the in-class worksheet on countability and cardinality. We will continue the discussion of the Schröder-Bernstein theorem briefly next time.
  3. Complete Exercises 1, 3, 5, 7, 9 from Chapter 2 of Rudin.
  4. Hand in Your proof of the Schröder-Bernstein theorem; Exercise 7 from Chapter 2. Due 09/26.
09/14
  1. Read the first section of Chapter 2 in Rudin (Finite, Countable, and Uncountable Sets). Send me a question on the reading by Monday evening. Due 09/18.
  2. Complete Exercises 3, 4, 6, 7, 8, 9, 11, 12, 17 from Chapter 1 of Rudin.
  3. Hand in Exercises 6, 8, 9. Due 09/19.
09/12
  1. Re-read Chapter 1 of Rudin carefully, and come to class prepared with any final(?) questions on this material.
09/07
  1. Read the remaining sections of Chapter 1 in Rudin (The Real Field, The Extended Real Number System, The Complex Field, and Euclidean Spaces). Send me a question on the reading by Monday evening. Due 09/11.
  2. Complete Exercise 7 of the in-class worksheet, on the ordered field of rational expressions.
  3. Hand in Exercises 4 and 7 of the in-class worksheet. Due 09/12.
09/05
  1. Read the first three sections of Chapter 1 in Rudin (Introduction, Ordered Sets, and Fields). As we saw in class, Rudin’s style is terse, so make sure you read carefully and understand every detail in his proofs. Send me a question on the reading by Wednesday evening. (Feel free also to let me know whether this is a reasonable amount of material, or too much—or too little.) Due 09/06.
  2. Work on Exercises 4 and 5 of today’s in-class worksheet, on the square roots of 3 and 4.