Math 63
Real Analysis
Last updated February 27, 2021 13:54:53 EST

General Information HW Assignments Canvas Page Lecture Videos


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Homework Assigments


Week of January 4 to January 8
Assignments Made on:
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
Assignments Made on:
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:
    1. 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 $\mathbf R$.)
    2. 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
Assignments Made on:
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
Assignments Made on:
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
Assignments Made on:
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
Assignments Made on:
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
Assignments Made on:
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
Assignments Made on:
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
Assignments Made on:
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 $\sum_k b_k$ be the series $$(a_1-\frac12 a_1-\frac12 a_1)+(a_2-\frac12a_2-\frac12a_2)+\cdots$$ More formally, $b_{3k+j}=a_k$ if $j=0$ and $-\frac12 a_k$ if $j=1$ or $j=2$.
Wednesday:
  • Study:
  • Do:
Friday:
  • Study:
  • Do:


Dana P. Williams
Last updated February 27, 2021 13:54:53 EST