Math 63
Real Analysis
Last updated June 27, 2016 13:25:50 EDT

Announcements:

• The final exam will be all take-home and will be passed out Friday. It will be due by noon on Wednesday, March 13th.
• In problem 5c of the Take home, the displayed equation should read $\sum (-1)^{n+1} a_n = a_1 -a_2 + a_3 - \dots$. (The $a_n$ was omitted.) And yes, I know we proved the AST in lecture. I want you to give a different proof use part (b).

### Homework Assigments

Week of January 7 - 11, 2013
(Due on Monday, January 14th)
 Monday: Study: Chapter I.1 to I.4 Do: Chapter I: 3ab, 5a, 7cf, 10ab Wednesday: Study: Chapter II.1 and II.2 Do:Chapter II: 2ab, 3, 6. Friday: Study: Chapter II.3 and II.4 Do: Chapter II: 10a, 11, 12, 13.

Week of January 14 - 18, 2013
(Due Tuesday, January 22)
 Monday: Study: Read III.1 Do:Chapter III: 1a Wednesday: Study:Read III.2 Do:Chapter III: 3, 4, 5. Friday: Study: Read III.3 Do:Chapter III: 8, 10, 11, 18. Reminder:No class Monday. Homework is due Tuesday in our x-hour.

Week of January 21 - 25, 2003
(Due Monday, January 28)
 Monday: NO CLASS: Think:"Rarely do we find men who willingly engage in hard, solid thinking. There is an almost universal quest for easy answers and half-baked solutions. Nothing pains some people more than having to think." -- Martin Luther King Jr. Tuesday (X-HOUR): Study:Read III.4 Do:Chapter III: 24 Comments: Here are some solutions to the first assignment: click here. Wednesday: Study: Read III.5 Do: Chapter III: 32 Friday: Study: Read III.5 Do: No additional assignments this week. Study for exam on Tuesday.

Week of January 28 - February 1
(Due WEDNESDAY, February 6)
 Monday: Study: Read section III.6 and start IV.1. Of course, you should also continue to prepare for the minor midterm on Tuesday. Do: In Chapter III: 38 X-Hour: IN CLASS EXAM: The in-class portion of the exam will be given during our x-hour. TAKE HOME: The take-home portion of the exam will be passed out Tuesday and will be due at the start of class on FRIDAY. You may access your text and class notes, but NO OTHER SOURCES are allowed annimate or inannimate. Wednesday: Study:Read section IV.1 (and yes, I know you also have a take-home exam.) Do:No written assignment because; because, yes, I know you also have a take home exam. Friday: Study:Read section IV.2 Do: In Chapter IV: 1d, 2, 4. By popular demand, this assignment is now due Wednesay. Note that Monday's assignment, once its posted, will also be due Wednesday. Here is a link to some solutions for the first exam. Here is a link to a hist-o-gram of the scores on the first exam. Please remember that the exam was out of a total of 75. While it is a bit early in the game to think seriously about grades, let's say that scores above 50 are "A-ish". Everything else is some flavor of "B".

Week of February 4 to 8, 2013
(Due Wednesday, February 13)
 Monday: Study: Read section IV.3 Do: In Chapter IV: 9ab and 10bc. Please note that this is due Wednesday (the 6th) in class. Tuesday (X-HOUR): Study: Read section IV.4 Do: In Chapter IV: 14ab (see Example 2 in section IV.1). Please note that his assignment is due a week from Tomorrow: Wednesday, February 13th. Wednesday: Study: Read section IV.5. Do: In Chapter IV: 29b. This assignment is due Wednesday, February 13th. There is no class Friday. Friday: Study: No Class Do: Enjoy Winter Carnival.

Week of February 11 - 15, 2013
(Due Wednesday, February 13 & 20)
 Monday: Study:Section IV.6 Do: In Chapter IV: 33 and 37. These problems are due Wednesday the 13th. Wednesday: Study: Section IV.6 Do: In Chapter IV: 42 and 43. This assignment is due Wednesday, February 20th -- that day after the in class portion of our second midterm. Friday: Study: Study Chapter IV and think about exam on Tuesday. Do: Show that if $f:E\to F$ is a continuous bijection between compact metric spaces, then the inverse $f^{-1}:F\to E$ is continuous. (Hint: why is it enough to show that $F(S)$ is closed if $S$ is closed in $E$?) Use the above to show that there is no continuous bijection from the unit interval $I=[0,1]$ to the unit square $I^2:=[0,1]\times [0,1]$. You are allowed to assume the validity of problem 29a in Chapter IV and that $I^2\setminus \{pt\}$ is arcwise connected. In class demos: animation. And the web page.

Week of February 18 to 22, 2013
(Due Wednesday, February 27)
 Monday: Study: Read sections V.1, V.2 and V.3 Do:Study for exam. X-HOUR: Midterm exam:In class portion. Take Home: Is due on Friday Wednesday: Study: Section V.4 Do:In Chapter V: 1b, 12. Friday: Study: Section VI.1 and VI.2 Do:In Chapter VI: 11. (In problem 11, you can assume that $f$ is Riemann integrable.) This was originally written as problem 11 in Chapter IV, but of course, there is no $f$ in that problem.

Week of February 25 - March 1, 2013
(Due Wednesday, February 27 and Wednesday, March 6)
 Monday: Study: Read VII.1. Do: In Chapter VI (Riemann Integration): problems 16 and 21. Let $s_n:= 1 +\frac1{1!}+\frac1{2!}+\cdots +\frac1{n!}$. Use Taylor's Theorem to show that for $n\ge 1$ we have $\frac 1{(n+1)!} < e-s_n <\frac e{(n+1)!}$, where $e=\exp(1)$. Use the above to show that $e$ is irrational. (Hint: suppose that $e=m/n$, and multiply the above inequalities by $n!$. Note that there is no integer $m$ such that $0< m <1$.) This assignment is due Wednesday. This is the last assignment that will be collected. Wednesday: Study: Read VII.2 Do: In Chapter VII: 8 and 9. Although homework is no longer collected, this material is fair game for the final. However, you can ask about any assigned homework even after the take home final has been passed out. Friday: Study: TBA Do: TBA