12/17/20, 4 p.m.: I have made a minor clarification, namely, an
example of a sequence Problem 1 on the Final online. Make sure to
download the latest version of the test.
12/17/20, 4 p.m.: I have made a minor clarification, namely, an
example of a sequence Problem 1 on the Final online. Make sure to
download the latest version of the test.
12/17/20: The Final Exam is posted on the Homework and Exams page.
12/16/20: Solutions of Midterm 2 are posted on the Homework and Exams
page. Refresh the page (or remove cache) in your browser, if you do
not see the solutions there.
12/16/20: I will be holding pre-final office hours tomorrow, Thursday,
3:30-4:30 p.m., and Friday, 11-noon. Sorry, at first I thought the
standard Thursday-Friday office hour schedule would be optimal, but I
had to change it.
12/16/20: The posted Class Notes from today's class meeting included a
correction, as suggested by Nhung, of those inequalities with which I
terribly messed up at the end of the class. Sorry! Now it all looks
clean and neat in the corrected Class Notes.
12/16/20: The mean on Midterm 2 was 39.63 (79%) out of 50. Two people
got the maximum possible score of 50. Most of you did a great job on
this test!
12/16/20: I have posted the problems from our problem sessions and
class notes from them, as well as from today's midterm discussion on
the Class Outlines page. I am working on posting solutions to Midterm
2 and writing the Final Exam, to be posted tomorrow afternoon.
12/10/20, 2:45 p.m.: I have made another clarification to Problem 5 on
Midterm Exam 2: the derivative of the function must be strictly
increasing.
12/10/20, 2:20 p.m.: I have made a minor correction to Problem 5 on
Midterm Exam 2: the function was supposed to be defined and continuous
on the closed inteval [a,b], rather than the open one, (a,b).
12/10/20: Midterm Exam 2 is posted on the Homework and Exams page.
12/9/20: I have noticed a type in Problem 8 right now. I am amazed
nobody pointed it out to me yet! Perhaps, most of you took it as an
obvious typo: the function is defined on [1,∞) and in Part (1)
you are asked to prove something about that function on
[0,∞). Of course, it should be [1,∞). Please correct that
in your homework! I apologize.
12/8/20: In addition to extra office hours today, I am moving my
Wednesday afternoon office hours tomorrow to the morning, 10:10-11:00,
just for that day, before HW 11 is due.
12/7/20: I will be holding extra office hours on Tuesday, December 8,
2:30-3:20 p.m., the day before HW 11 is due.
12/4/20: I have corrected errors in Problems 6 and 8 on Homework
11. Sorry!
12/2/20: Homework 11 is posted.
12/2/20: I will be posting HW 11 tonight.
12/1/20: Coverage for Midterm II, December 11: In short, all we
will have studied roughly between the two midterms, from the 10/14
class meeting (monotonic sequences) through the 12/04 class meeting
(the contraction mapping theorem). See the class notes on our Class
Outlines page, as well as Rudin, Sections 3.13-3.26, 3.28, 3.30-3.37
(skipping the proof of 3.37), 3.41-3.43, 3.45-3.47, 3.52-3.55, Chapter
4 (skipping 4.17), Chapter 5 (through 5.15), plus the
higher-derivative test for relative extrema (not covered by Rudin, see
class slides) and the contraction mapping theorem and Newton's method
(9.22-9.23 and Exercises 5.22, 5.25).
12/1/20: In addition to extra office hours today, I am moving my
Wednesday afternoon office hours tomorrow to the morning, 10:10-11:00,
just for that day, before HW 10 is due.
11/30/20: There will be extra office hours on Tuesday, December 1,
2:30-3:20 p.m., the day before HW 10 is due.
11/21/20: Homework 10 is posted.
11/20/20: It turns out we have been moving faster than I thought. And
I know, many of you will be moving back home during the Thanksgiving
Break and could be busy with other things. So, I am putting off the
due date for the next homework till Wednesday, December 2. This
homework will be posted on Saturday, November 21.
11/16/20: As I told you at the beginning of today's class, I plan to
make the remaining exams (Midterm II and the Final) shorter and allow
shorter time.
11/14/20: Having looked at the new Homework 9 with a fresh eye in the
morning, I have slightly changed some problems.
11/14/20: Homework 9 is posted.
11/11/20: Forgot to mention today in class that uniform continuity on
a domain is stronger than continuity at each point of the domain,
sometimes called pointwise continuity. Indeed a uniformly
continuous function is continuous at each point a of the
domain: just plug y = a in the definition of uniform
continuity, and it will turn into the definition of continuity
at a.
11/7/20: Homework 8 is updated (to 7 problems).
11/7/20: Homework 8 is posted.
11/3/20: Our grader Tom Winckelman has written an answer key to
Problem 6 on Homework 6, which I have just posted on our Homework
page.
11/2/20: I have posted the slides from today's class meeting. After
the class, I have realized that there was no need to pass to the limit
as n → ∞ in the proof of the rearrangement theorem for
absolutely convergent series and simplified the argument a bit on the
slides. What I presented to you in class was correct, but that step
could be avoided. Sometimes we tend to oversmart ourselves...
10/31/20: Homework 7 is posted.
10/31/20: I have posted a solution, with explanations on how to arrive
at it, for Problem 1 on Homework 6 on our Homework page.
10/28/20: The way I worded the divergence statement of the Ratio Test
today is a little weaker than that in the textbook. My condition for
divergence was lim inf |ak+1/ak| > 1, whereas
the textbook proves that if |ak+1/ak| ≥ 1
for k ≥ N for some N, then the series ∑ ak
diverges. The condition lim inf |ak+1/ak| > 1
implies Rudin's condition |ak+1/ak| ≥ 1 for
k ≥ N for some N, because if lim inf
|ak+1/ak| = L > 1, then for some ε > 0
and N ≥ 1, we will have |ak+1/ak| > L -
ε > 1 for all k ≥ N. This is why Rudin's wording of the
Ratio divergence test is stronger and Rudin's proof of the test also
proves the Ratio divergence test as worded in class. I left the Ratio
Test for your own reading, and the proof of it is very much like that
of the Root Test, but even simpler. (Not surprising, because the Ratio
Test is weaker, as we see from those inequalities in Section 3.37 or
on the last slide of today's class notes.) The two examples in the
textbook also illustrate that the root test is stronger and that the
condition lim sup |ak+1/ak| > 1 does not imply
divergence, which explains why the Ratio divergence test looks so
different from the Root divergence test.
10/28/20: I have removed Problem 7 from HW 6, as I have realized it is
not covered by the material we have studied so far. Already on Monday
I realized we would not get to summation by parts, and I removed
Sections 3.41-3.42 from the reading part of the assignment, but I
forgot that Problem 7 was actually on summation by parts. I will
include this problem in the next problem set, though.
10/27/20: Solutions of Midterm 1 are posted on the Homework and Exams
page. Refresh the page (or remove cache) in your browser, if you do
not see the solutions there.
10/26/20: I have posted today's class slides. Before posting, I
brushed the notes up a little bit. In particular, I left the fact that
every subsequential limit of a sequence {ak} will be
between its lim inf and lim sup as a simple observation, given that
all the terms ak for k ≥ m will be between
lm and um (therefore, all but finitely many
terms of {ak} will be outside of each interval
[lm, um]). And I have added another simple
observation that lim inf ak = lim sup ak = L iff
lim ak = L, which I will return to next time.
10/26/20: I am working on posting short solutions of Midterm 1.
10/26/20: On Homework 6, I have changed Problem 5, the problem on
limits superior and inferior, so as to assume the sequence is
bounded. The statement is true in the unbounded case, but the
unbounded below and above cases should be treated separately, like in
Problem 6.
10/25/20, 2:30 a.m.: Homework 6 is posted.
10/24/20: Only now I have graded your midterms. It was one of the
hardest grading jobs on my life! You, guys, wrote so many various, at
times, cryptic, arguments! I was impressed with seeing so many
ideas. Not all of them were to the point, but it was still interesting
to see a lot of unconventional wisdom. I do apologize for the delay in
grading. The average score is 50.27 out of 70, and it corresponds to
72%. I graded your exams quite carefully and rather harshly, but the
average score has turned out to be quite high for such a hard class. I
am moving on to creating the homework, which is also overdue.
10/21/20: I have improved the wording of Problem 1 on Homework
5. Reload to see the latest version of the pdf file.
10/21/20: I have added that extra slide which shows why e :=
lim (1+1/n)n ≥ 1 + 1/1! + 1/2! + ... to the slides of
today's class. See our Class Outlines
page.
10/18/20: I have made a few changes to Homework 5: defined discrete
spaces, changed Problem 8 upside down, and made some other minor
changes.
10/17/20, 1 a.m.: Homework 5 is posted.
10/16/20: I am reminding you that there will be no class meeting
today. You may finish working on your exams during the class time,
and I will start grading...
10/15/20: Some of you are still puzzled about what is asked for in
Problem 7 on Midterm 1. "Exactly those" is an "if and only if"
statement. It means show that the intervals contain all their
intermediate points and every set which contains all its intermediate
points must be an interval.
10/14/20: After my today's office hours, I have added a clarification
on what is meant by an interval in Problem 7 on Midterm 1, as well as
clarified what results may be used on the exam (see the Rules section
of the exam). I did not think these clarifications were necessary,
but since some people were wondering, I thought, why not? Reload the
exam page to get an updated copy of the exam.
10/13/20: I have added a clarification of what it means to fix
ℝ in the problem on automorphisms of ℂ on Midterm 1.
10/10/20: The take-home Midterm 1 is posted on the Homework and Exams
page of this web site. If you do not see it there, reload the
page. Good luck!
10/8/20: One question has come up in regards to Problem 1 on HW 4. You
are not supposed to use the theorem that the Bolzano-Weierstrass
property implies the compactness property. The matter is that you will
be proving that B-W implies compactness on the take-home midterm, and
this homework problem (or rather, your solution of it) may be used as
a step to proving that theorem B-W=> compactness on the exam.
Rudin's Theorem 2.41, which does discuss these things, is about X = ℝn only. The proof of (a) => (b), used there to show (c)=>(b), applies only to ℝn. Problem 1 on the homework is about a general metric space.
10/8/20: I have update the homework, so as to add a hint to Problem 5:
you may take it for granted that lim (2/3)k = 0.
10/7/20: If you have seen that I wrote earlier that I would be posting
the take-home Midterm Exam 1 before tomorrow morning, it was an error
I have corrected below. I had a sleepless night, because of a grant
submission deadline (Professors also have due dates for their
homework!), submitted it, and was so happy that it felt like "Thank
God it's Friday!" :-)
10/7/20: I will be posting the take-home Midterm Exam 1 before
Saturday morning. Unlike working on your homework, no study groups or
cooperation when doing the exam! You may use any textbooks and
internet sources, but just copying a solution you might occasionally
find will not gain any credit and will be regarded as plagiarism. You
have to present all solutions in your own words.
10/7/20: If you do not see the version of HW 4 with simplified Problem
7, see below, just refresh the Homework 4 page.
10/7/20: During today's office hours, I have realized that we do not
have good means (except convexity) to prove that a reasonable
connected subset of ℝ2 is indeed
connected. Therefore, I am changing Problem 7, so as you do not have
to explain why the subset you have come up with is connected. Just
make a right guess. Good intuition is based on the fact that a
path-connected subset of ℝ2 has to be
connected. However, the notion of path connectedness is based on
continuous functions which we have not studied yet, so it is illegal
to use it at this juncture. However, we do have means to show that a
disconnected set is disconnected. Thus, you will need to explain why
the interior of the subset you have presented is disconnected.
10/3/20: I have made a terrible mistake on Friday, answering Juliana's
question about nested sequences of open intervals. They may have empty
intersection, unlike nested sequences of closed intervals. I have
repaired this on the current homework by asking you to construct a
counterexample.
10/3/20, 2 a.m.: Homework 4 is posted.
10/1/20: I have realized that I have cheated on Homework 3: I put in
two Problems #6, so as the homework appears to be made of 8 problems,
while there are 9 of them. I will correct the numbering in a few
minutes. (I should have used a bijection, but shamelessly violated the
pigeonhole principle instead!) During office hours, a correction of
the last problem came up: it should be "at most countable subset"
there in the definition of a separable metric space. I am correcting
this, too.
9/26/20: Homework 3 is posted.
9/19/20: Homework 2 is posted.
9/18/20: Again, I have posted today's slide presentation. See
Class Outlines. I will be posting HW 2 much
later tonight.
9/17/20: Please upload homework to the Math 5615H Canvas page by 12:20
p.m. on Friday. If you do not know how to do that or have trouble
doing that, let me know.
9/16/20: Our grader Tom Winckelman has made available a useful
resource: Analysis
Proofs. (Once there, click on PDF in the upper left corner to
see the text.) This text is designed as a companion to baby Rudin,
providing more detailed and probably more accessible proofs to
statements in baby Rudin. It will also allow you to receive extra
credit: 0.5% for spotting a math typo or error in Tom's companion.
9/16/20: See the Class Outlines page to
find a link to today's slide presentation.
9/14/20: Good news: we have got a grader. This means there will be no
quizzes, but homework, two midterms, and a final. And the first
homework is due before the beginning of this Friday's class, at 12:20
pm, September 18. So far, the submission mode is uploading your
homework to Canvas. Here are some ideas on how to scan your work,
unless you do not use TeX or LaTeX for creating
homework: https://gradescope-static-assets.s3-us-west-2.amazonaws.com/help/submitting_hw_guide.pdf. Poor
scanning should be avoided: be compassionate to the grader and me.
9/14/20: I have updated links from our class web page to our Library
Course Page, which contains a link to the solution manual to Rudin's
exercises. I have also updated the Syllabus to reflect the 1-point
class participation policy and include a link to a list of private
tutors maintained by Math Ph.D. students.
9/11/20: Homework 1 is posted.
9/11/20: I love discussions with you folks, but today's lively
discussion was a bit overwhelming, and I was able to present only
about 1/3 of what I wanted to discuss. I am now imposing a 1 point cap
for class participation per class. This does not mean that your should
shut up, when you get your point. :-) Let us see how it works. I am
open to suggestions of a more efficient system of incentives to get
the class to participate.
9/11/20: I will be posting homework about a week before it is due. The
first homework will be posted tonight and will be due next Friday,
9/18. If we do not get the grader, I will not grade it, but rather
give you a quiz on the due date.
9/9/20: Hinted by today's discussion of class policies, I am imposing
a cap of 2 points for each class participation. I have also posted
your scores on Canvas. Let me know if you do not see you score. So
far, given that I have designed a pretty straightforward class web
page, I plan to use Canvas only for posting your grades, such as your
class participation score.
9/9/20: Please, stop by during my office hours -- you must have
received Zoom links via Google Calendar invitations from me. Visit my
office hours, even if you do not have questions. You may just say hi,
introduce yourself, chat with me about anything, etc. Otherwise, it
feels lonely to be sitting there in the internet wilderness all by
myself. Stopping by is the easiest way to break me out of solitude and
make my day!
9/9/20: I have changed my Wednesday office hours from 11:15-12:05 to
1:25-2:15 to make myself available after today's class and all
Wednesday classes in the future. I have also changed my Thursday
office hours to 2:30-3:20 to be able to participate in the School of
Mathematics Colloquia.
9/9/20: I recommend the following way to study for this class. Attend
each online class meeting, take notes, participate in class actively
(it is part of your grade). Keeping your video on is a requirement for
Zoom class meetings. If you stop your video for more than 10 seconds
with no warning, I may remove you from the Zoom meeting, and you will
have to rejoin. If you start checking your email or doing random
things on your computer in class, you will see how hard it would be to
follow the class afterward. A good way to prevent yourself from being
distracted by the virtual world is to take notes on a piece of
paper. After each class, review your notes and study the corresponding
part of the text. You can find out which part of the text at
the
Class Outlines page. Then do the assigned homework problems
pertinent to that material. Some students find it helpful to read the
material before it is covered in class, some prefer to do reading
after class.
9/9/20: If you are an undergraduate looking to register for this
class, you need the instructor's and an academic advisor's
approval. If you are a graduate student, you just need the
instructor's approval. However, this section is full, and I will not
be giving approvals, until Section 3, which meets at the same time, is
full.
Last modified: (2020-12-17 16:05:55 CST)