Math 63
Real Analysis
Last updated March 07, 2021 15:36:13 EST

Announcements:

• Our first homework assignment will be due Friday the 15th via gradescope.
• Because of the pandemic, it is possible, even likely, that course details such as exam dates may evolve. It is your responsibility to keep apprised of any changes during the course of the term.

### Homework Assigments

Week of January 4 to January 8
 Monday: Study: Do: Wednesday: Study: Do: Friday: Practice Exam and Survey: I have created a survery in the form of a practice exam on gradescope. You should login to gradescope, download "m63-exam-survey" and complete it (you'll have two hours). Then scan, upload, and link your responses. Please do this by Monday evening. Study: Read all of Chapter I in the text. Do:In Chapter I, work: 3ab, 5b, 7bd, and 10ab.

Week of January 11 to 15
 Monday: Study: Read sections II.1 and II.2. We will not be able to finish all of II.2 today. Note that the results in these sections hold for any ordered field. Do: Show that in $\mathbf R$, if $x\ge -1$ and $n\in \mathbf N$, then $(1+x)^n\ge 1+nx$. I suggest using induction. (You can work this problem in any ordered field if we define $(n+1)\cdot x= n\cdot x + x$ for $n\in\mathbf N$. But it is fine to work in $\mathbf R$.) In Chapter II: 2 and 3. Wednesday: Study: Read Sections II.3 and II.4. (Note: I first posted Friday's assignment here an missed out this assignment completely. I am sorry for any confusion.) Do:In Chapter II: 6, 11, and 13. I found problem 1 from the previous assignment helpful in working question 11. Friday: Study: Read section III.1. Observe that there is no class on Monday (January 18th) due to the MLK holiday. Do: In Chapter III: 1a and 2.

Week of January 18 to 22
 Monday: Study: No Class: MLK Day Do: Wednesday: Study: Read Section III.2 Do: In Chapter III: 3, 4, and 5. Thursday (x-hour): Homework Solutions: Here are selected solutions for the homework. (Last modified January 21, 2021) Homework: The assignments for Friday and Wednesday will be due tomorrow (Friday, January 22). Todays assignment will be due next Friday. Study: Read Section III.3 Do: In Chapter III: 10 and 11. Friday: Study: Read Section III.4 Do: In Chapter III: 24.

Week of January 25 to 29
 Monday: Study: Read Section III.5. We will finish III.5 on Wedneday Do: In Chapter III: 28 and 32. Wednesday: Study: Finish III.5 and III.6 Do: No assignment today. Friday: Study: Read Section IV.1 Do: In Chapter IV: 2 and 3. (What about the converse to the assertion in 2?)

Week of February 1 to 5
 Monday: Study: Read Section IV.2 and Start Section IV.3 Do: No new assignment. Wednesday: Study: Finish Section IV.3 and start Section IV.4 Do: In Chapter IV: 9ab and 10bc. Friday: Study: Finish IV.4 Do: In Chapter IV: 14, 16, and 17 (just for $f(x)=\sqrt{|x|}$).

Week of February 8 to 12
 Monday: Study: Read Section IV.5 Do: In Chapter IV: 26 and 29b. (This was previously 29a which we proved in lecture. Please do 29b instead and don't turn in 29a.) Wednesday: Study: Start Section IV.6 through page 87, Do: In Chapter IV: 33b and 37. Friday: Study: Finish Section IV.6 Do: In Chapter IV: 41, 42, and 43. Note that in problem 41, it is assumed that the sequence $(f_n)$ converges to a continuous function $f$.

Week of February 15 to 19
 Monday: Study: Read Sections V.1 to V.4 Do: In Chapter V: 1c, 2, and 6. Wednesday: Study: Read Sections VI.1 and VI.2 Do: In Chapter VI: 2. Friday: Study: Read Section VI.3 Do: No assignment for today.

Week of February 22 to 26
 Monday: Study: VI.4 Do: In Chapter VI: 8, 11, 17, and 20. Wednesday: Study: Read Section VI.5. Do: In Chapter VI: 16, 17, 21. I had previously assigned 26 but I now don't believe we have the tools to do it, so please do not turn it in. Friday: Study: Read VII.1 Do: In Chapter VII: 2 and 8a.

Week of March 1 to March 5
 Monday: Study: We're exploring Section VII.2 Do: In Chapter VII: 9 and 12. Find an example of a convergent series $\sum_{k=1}^\infty b_k$ such that $\sum_{k=1}^\infty b_{k}^3$ diverges. I suggest the following (shown to me indepdently by Peter Doyle and George Welch). Let $(a_n)$ be a sequence of positive numbers converging monotocally to $0$ such that $\sum_{n=1}^\infty a_{n}^3$ diverges. Then let $\sum_{k=1}^\infty b_k$ be the series $$(a_1-\frac12 a_1-\frac12 a_1)+(a_2-\frac12a_2-\frac12a_2)+\cdots$$ More formally, for $k\ge1$ we have $b_{3(k-1)+j}=a_{k}$ if $j=0$ and $-\frac12 a_{k}$ if $j=1$ or $j=2$. Wednesday: Study: Read Section VIII.3 Do: In Chapter VII: 22 and 24. Show that if $f$ is a differentiable function on an open interval $U\subset \mathbf R$ such that $f'=f$, then $f(x)=Ce^x$ for some real constant $C$. (I suggest considering $h=e^{-x}f$.) Conclude that $e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$ for all $x\in \mathbf R$. Friday: Study: We're still in (or at least near) VIII.3. Do: No new written assignment today. There may be additional homework on Monday and Wednesday next week, but no more homework will be collected after today. For your own amusement, you may want to convince yourself that $\int_0^\infty e^{-t} t^n\,dt=n!$ for $n\in \mathbf N$. 