12/11/14:
Sample Final, Part II, is posted completely.
12/11/14:
Sample Final, Part II, is posted completely.
12/10/14: Sample Final, Part II, is partially posted. More problems to
follow.
12/10/14: As I have told you, I will have special office hours during
the Study Period. Here they are: Thursday: 10:00-11:20, Friday:
12:00-12:50, Monday: 10:00-10:50. I am also available by
appointment, as usual.
12/10/14: Now
the
Sample Final, Part I, is posted with solutions to Problems 4(1),
4(2), and 5.
12/8/14: The mean on the second Midterm is 22.31 out of 40 or
55.775%. The median is 23.5. This is the statistics within my
two sections 001 and 003 of Math 5615H. As I told you earlier, I will
be assigning grades at the end of the term based on a curve across
both sections.
12/07/14:
A
Sample Final (Part I) is posted.
12/03/14: Reminder: The students currently taking Math 5615H with
Instructor X have a priority to register for the section of
Math 5616H next term with the same instructor through December
5. After that it will be on a first-come, first-served basis. Online
registration is currently closed. This does not necessarily mean that
you cannot register. To register, send a message to
ugrad@math.umn.edu, give them a call at (612) 625-4848, or stop by in
Vincent Hall 115 to request a permission number.
12/3/14: Now a solution to Problem 4 is posted within
the
Sample Test II.
11/30/14:
A
Sample Midterm II is posted.
11/26/14: Also, I have added a hint to Problem 2 on HW 11: Use
induction on n and change of variables y = 1/h
before applying L'Hôpital's rule.
11/26/14: I have made cosmetic changes to HW 11: corrected an obvious
typo and clarified: “under what conditions on n” in
Problem in 4(2).
11/24/14: The students currently taking Math 5615H with Instructor X
have a priority to register for the section of Math 5616H next
term with the same instructor through December 5. After that it will
be on a first-come, first-served basis. In any case, if you are not
able to register online, send a message to ugrad@math.umn.edu, give
them a call at (612) 625-4848, or stop by in Vincent Hall 115 to
request a permission number.
11/24/14: Made another correction in the same Problem 6 on Homework
11. This does not mean the problem is going to be undoable.:-) Try to
use the Mean Value Theorem iteratively.
11/23/14: Made a correction in Problem 6 on Homework 11.
11/22/14: Homework 11 is now complete.
11/22/14: Part I of new homework, due Monday after Thanksgiving, is
now posted. Sorry for the delay!
11/22/14: Coverage for Midterm II was posted here on 10/14, see
below. Here is coverage for the Tuesday, December 16, Final
Exam (8:00–10:00 a.m. (Section 001) and 1:30-3:30 p.m. (Section
003) in your regular classroom): Chapters 1-5. How to get ready for
the test? The best way would be to try to solve as many problems
as you can on your own (with an open textbook, if needed, and no
timing) from the sample final exam (to be posted sometime after
Thanksgiving), the homeworks, and the textbook. For those problems you
cannot solve, it may be a good idea to discuss them with your friends,
i.e., your study group. Come to my office hours to get hints or other
help. Go over homework problems again without consulting your
solutions and look for problems from exams in similar classes on the
web: search for Analysis tests, exams, midterms, finals, etc.,
but make sure the class you find is for undergraduates. When you know
you can get an idea of how you might approach most of the problems
right away, you may feel more sure that you will do well on our
exam. Good luck!
10/19/14: I have added another hint to HW 10, now for Problem 5(2).
10/19/14: I have added a hint to Problem 8 and a clarification to
Problem 2 on HW 10.
10/17/14: For the current homework, HW #10, you may assume that
(√ x )'
=
1/(2√ x ),
sin' x = cos x and similar computations known to you from calculus.
10/15/14: New homework is now posted. Sorry for the delay!
10/14/14: Solution to Problem 6(2) for Homework 8 is posted. Working
on the new problem set now.
10/14/14: Coverage for the Friday, December 5, Midterm
Exam (in your regular classroom) is basically all we will have
studied since our first midterm through Wednesday, December 3. This
will be Chapters 3-5. How to get ready for the test? The best
way would be to try to solve as many problems as you can on your own
(with an open textbook, if needed, and no timing) from the sample exam
(to be posted sometime during the Thanksgiving break), the homeworks,
and the textbook. For those problems you cannot solve, it may be a
good idea to discuss them with your friends, i.e., your study
group. Come to my office hours to get hints or other help. Go over
homework problems again without consulting your solutions and look for
problems from exams in similar classes on the web: search for Analysis
tests, exams, midterms, etc., but make sure the class you find is for
undergraduates. When you know you can get an idea of how you might
approach most of the problems right away, you may feel more sure that
you will do well on our exam. Good luck!
11/12/14: I have added a correction to Problem 9 on Homework 9: you
should assume that f: X ----> X' is continuous.
11/8/14: I have posted Homework 9 only now, because I had trouble
connecting remotely from my home to the departmental server last
night. Sorry! Enjoy it, though! I think this homework is simpler than
the previous two homeworks.
11/6/14: Some clarifications and hints for HW problems.
Problem 1(4): Group terms and estimate sizes, as we did that for Problem 5 on the previous HW. Or just notice whether the three series \sum 1/(4n+1)2, \sum 1/(4n+3)2, \sum 1/(2n+2) converge or not and conclude something about a subsequence of partial sums of the given series.
Problem 4: We assume in p/q the numerator p is integral and the denominator q is natural.
Problem 6: Do it using the epsilon-delta definition. Also, try to guess how to estimate |d(p,q) - d(p_0,q)| by d(p,p_0) and prove it, using the axioms of a metric space.
11/6/14: In the announcement of 10/23 about rescheduling my Monday
office hours this week, I made an error: it should be my Monday,
November 3, office hours are rescheduled to Thursday, November 6,
2:20-3:10 p.m. Sorry, somebody has just pointed that out to me. In
my announcements of that in class, I did mention this Thursday, so
hopefully, most of you assumed the rescheduled office hours would be
today. As always, feel free to make an appointment for another time or
use e-mail to get in touch with me.
11/4/14: I have realized that I need to correct my answer to a
question asked by Shashank the other day in my 12:20 p.m. class. I do
not think it is true that if a subsequence of partial sums of a series
converges and the sequence of terms of the series tends to 0, then the
series converges. You, guys, may think of a counterexample. If you
find one, make sure to tell me about it! However, what is true (and
that is what we have used so far) is the following: if a subsequence
of partial sums of the type s_{Mn+N}, where n runs over the naturals,
and M and N are fixed naturals, converges and the sequence of terms of
the series tends to 0, then the series converges. I sketched a proof
of that for M = 2 and N =1 yesterday in class. The general case is
done the same way.
11/1/14: Made some changes to HW 8. Make sure you see a set with 8
problems on two pages.
11/1/14: Homework 8 is posted. Enjoy!
10/30/14: Corrected the hint to Problem 5 on HW 7. Sorry!
10/29/14: Corrected the hint to Problem 3(5) on HW 7 and added a few
more hints.
10/27/14: Added a hint to Problem 3(5) on HW 7.
10/27/14: For Problem 1 on HW 7, I have realized that the text had an
estimate (1 + 1/n)n ≤ e in the proof of Theorem
3.31, but did not have an estimate like exp(1/(n+1)) ≤ 1 + 1/n,
which may be used to evaluate the limit. This estimate is easy to
prove similarly to (16) on p. 65 of the text: exp(1/(n+1)) = 1 +
1/(n+1) + 1/2!(n+1)2 + ... ≤ 1 + 1/(n+1) +
1/(n+1)2 + ... = (n+1)/n. So, I have added the phrase
“You can use L'Hôpital's rule” to the problem. Now
you may do the problem either way.
10/25/14: Homework 7 is posted. Sorry about a delay, but I have tried
to make it less time consuming than the previous one. Enjoy!
10/25/14: I have improved the posted solution to Problem 9 by
inserting a few comments. I am working on the homework now. Sorry
about the delay.
10/25/14: Now
a
solution to Problem 9 (HW6) is posted.
10/23/14: I have to reschedule my Monday, November 3, office hours
to Thursday, November 6, 2:20-3:10 p.m. If that is not good for
you, please make an appointment for another time or use e-mail to get
in touch with me.
10/21/14: Corrected to a typo in Problem 6: it must be
(1+h)n ≥ 1+ nh.
10/18/14: Complete new homework has been posted. Enjoy!
10/18/14: On Saturday, November 8, Marquette University in Milwaukee,
WI, will hold an open house for students interested in pursuing
graduate study in the Natural or Computational Sciences. Find out more
and register
at
http://www.marquette.edu/biology/open-house.php.
10/18/14: New homework has been posted, so far partially, due Friday,
as usual. Enjoy!
10/15/14: The mean on the first Midterm is 24.65 out of 40 or
61.625%, which is normal and slightly on the high side for this
class. This is the mean in my two sections 001 and 003 of Math
5615H. By the way, I will be assigning grades at the end of the term
based on a curve across both sections.
10/12/14: Typo corrected in homework Problem 5. Sorry! Thanks to those
who noticed and let me know.
10/11/14: New Homework is posted, due Friday, October 17. Enjoy!
10/8/14: Now a solution to Problem 3 is posted within
the
sample test.
10/6/14: I would like to remind you that guidelines on how to get
ready for the upcoming test were posted here on October 1. Just scroll
down to see them.
10/4/14, 1:40 a.m.:
A
Suggested Problems on the Material Covered in class October 1-8
are posted.
10/4/14, 1:20 a.m.:
A
Sample Midterm I is posted.
10/3/14: A
solution to Problem 2 on Homework 4 is posted. All the suggestions
on the solution I received from Jonathan and Joseph, as well as the
statements I made at the blackboard, in the 9:05 class on Friday were
correct. As Jonathan pointed out, we needed to articulate the argument
better. The solution I posted is one way of doing it.
10/1/14: Hint to Problem 2: First, construct a countable base
for A, that is, a countable collection {Ui}, i
∈ N, of open sets in X, such that if V is an open set in
X and x is in A ∩ V, then there exists some i such that x
∈ Ui ⊂ V. To construct such a base, show for
each n in N, there is a finite number of balls of radius 1/n
covering A. Use the constructed base to select a countable (or, if you
are lucky, finite) subcovering of a given open covering
{Vα}{α ∈ I} of A.
10/1/14: Coverage for the Friday, October 10 Midterm
Exam (in your regular classroom) is basically all we will have
studied through Wednesday, October 8. This is Chapters 1-2 (skipping
the Appendix to Chapter 1). How to get ready for the test? The
best way would be to try to solve as many problems as you can on your
own (with an open textbook, if needed, and no timing) from the sample
exam (to be posted this week), the homeworks, and the textbook. For
those problems you cannot solve, it may be a good idea to discuss them
with your friends, i.e., your study group. Come to my office hours to
get hints or other help. Go over homework problems again without
consulting your solutions and look for problems from exams in similar
classes on the web: search for Analysis tests, exams, midterms, etc.,
but make sure the class you find is for undergraduates. When you know
you can get an idea of how you might approach most of the problems
right away, you may feel more sure that you will do well on our
exam. Good luck!
10/1/14: Somebody has pointed out to me that in Problem 2, a
subcovering that you construct may turn out to be finite. I have
corrected that problem to require the existence of a countable or
finite subcovering.
10/1/14: Clarification: the Heine-Borel theorem for arbitrary metric
spaces says that a subset of a complete metric space is compact
if and only if it is closed and totally bounded, i.e.,
contained in the union of finitely many open balls of any given
size. However, we study only the Heine-Borel theorem
in Rn, which is simpler (and stronger): a subset
in Rn is compact iff it is closed and bounded.
9/29/14: If I did not mention clearly today in class that the fact
that the closure of an open ball of radius r was the closed ball of
radius r was true in Rk, here we go. Try to think of
a metric space in which this statement would be false.
9/28/14, 2:20 p.m.: Completed Problem Set #4. Sorry, it took me that
long to complete it. Make sure that the one you download has 9
problems. In the original version, Problem 5 was identical to Problem
4. In the final version, I tried to stick to the original numbering as
much as possible, i.e., replaced Problem 5 with a different problem
and kept the numbering of the rest.
9/27/14, 11:40 a.m.: Just six problems so far. I have to take a break
for a few hours. I will continue later. You may start working on the
six problems.
9/27/14, 1 a.m.: Just two problems so far. I am so tired that cannot
think straight. Have to go to bed. I will continue tomorrow
morning.
9/26/14: I have made a couple of errors in different classes
today. Apparently, it was not my day. :-) In the 9:05 class, I
answered the question about the cardinality of RN
incorrectly. Actually, the cardinality of this set is c, the
same as that of R. If you are interested in how such statements
are proven, you may take a look at the book on Set Theory by
Abhijit Dasgupta, available electronically through the library, or any
other set theory text.
The other mistake I made was in the 12:20 class. I sketched the graph of tan x, but called it arctan x. Respectively, the shift and dilation I applied were incorrect. A correct bijection from (0,1) to R would be: y = tan π(x-1/2). I should definitely brush up on my trigonometry! :-) Have a great weekend and look for the next homework sometime late tonight.
9/24/14: I have just realized that Problem 4 on HW 3 is pretty much
impossible to do without the Schroeder-Bernstein theorem, which we did
not cover. So, you can assume that the union in Problem 4 is the
disjoint union, in which case it is not hard to do this problem
without the Schroeder-Bernstein theorem. I apologize about the
confusion: just thought it could be done quite easily without the
disjointness assumption.
9/20/14, 12:33 a.m.: Just posted Homework 3. Enjoy!
9/19/14: I might have given wrong initials for Paul J. Cohen, who
proved the Continuum Hypothesis in 1963 and received a Fields Medal in
1966 (in Moscow ;-)) for that proof. (BTW, this shows that proofs are
important! ☺) He proved that the existence of a cardinal
strictly between ℵ0 = |N| and c =
|R| was independent of the standard axioms of set theory, i.e.,
we can add the axiom that there are no sets of intermediate
cardinality between ℵ0 and c or we can add the
axiom that there exists a set of such cardinality -- either way we get
as consistent a set theory as the set theory without this extra axiom
is, i.e., everything we are used to when working with sets will still
be there.
9/17/14: If you have a question to the grader, you may send a message
to him at wang2625@umn.edu. If you need to see him, make an
appointment. His name is Yuxiang Wang.
9/17/14: I have added a few hints
to
Homework 2. Reload or even remove your computer's cache and then
reload, if you do not see an update dated 09/17.
9/16/14: I have added a link to the brand new library course
page
https://www.lib.umn.edu/course/MATH/5615H for Math 5615H to our
course page and corrected it on the on-line syllabus. The old link was
missing an H in the URL address. The library page contains a link to
complete solutions to Rudin's exercises.
9/14/14: Changed HW 2 a little - sorry: moved the challenge problem,
#6, to the end; now it is Problem 12. And Problem 6 is now to show
that every automorphism of the reals is trivial.
9/13/14, 10:33 a.m.: Posted HW 2. Enjoy it!
9/13/14, 9 a.m.: Still working on the homework. Expect it any time
soon.
9/12/14: Somebody pointed a flaw in my argument in class today for
Problem A', which was, given a positive real x such that x2
> 12, find a rational y, such that 0 < y < x, but still y2
> 12. The flaw was that I wanted to make sure that h <
(12-x2)/(2x-h), but said choosing h: 0 < h <
(12-x2)/x would guarantee that. That logic was skewed,
because for h > 0 small enough, we would have
(12-x2)/(2x-h) < (12-x2)/x, as 2x - h > x >
0. What I needed to do was to take h: 0 < h < (12-x2)/(2x),
and then, since (12-x2)/(2x) <
(12-x2)/(2x-h). Thus the right choice for h in the end
could be h = min (x, (12-x2)/(4x)). A few points off my
homework score -- sorry.
9/11/14: I have added a paragraph of explanation on what
Problems A and B of Homework 1 are about, which is also
some sort of a hint. I have also added the condition x > 0 to
the set in Problem B. It does not change the solution much, so
if you have already solved the problem not assuming x > 0, then
just leave it as is. And if you have not solved it yet, then the
positivity condition should not make your life much simpler or harder:
just one little thing less to worry about.
9/10/14: My return flight to Minneapolis got delayed because of a
technical problem with the plane. I have just landed at the MSP
Airport and am heading to my office. I apologize about being late for
my office hours. I will be holding them today, Wednesday, from 2:50 to
4 p.m. and tomorrow, Thursday, from 2 to 3 p.m.
9/8/14: I have made small changes to Homework 1: wrote "skim the
Appendix" instead of "read the Appendix" and changed the phrase
"sufficiently small rational h" to "sufficiently small h" in the hint
to Problem A. Otherwise, the hint was confusing.
9/4/14: I have changed extra problems in Homework 1: split former
Problem A into two problems. Please, make sure you have Problems A, B,
C, when you start doing your homework.
9/3/14: I will be out of town, visiting Purdue University part of next
week this week, from Tuesday to Wednesday morning, September 9-10. I
have asked Professor Conn to substitute for me in my 9:05 Wednesday
class. If my flight back is late, he will also cover my 12:20 class on
Wednesday. I will hold my Wednesday, September 10, office hours from
1:25 to 2:15 p.m. instead of the regular time of 11:15 to 12:05.
9/3/14: I recommend the following way to study for this class. Attend
each class, take notes, participate in class actively. 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.
9/3/14: If you need to register for this class, please, send a message
to Ms. Lawson at ugrad@math.umn.edu and ask for a permission number. If
permission is granted, go to OneStop and register.
9/3/14: I have posted the
first
Homework, due Friday, September 12. It will be collected or
graded. Making sure to do homework on your own will be the best way to
get ready for the exams. Getting together with other students (that
is, in study groups) is a very effective way to do homework. Your
friend is your TA.
Last modified: (2015-09-21 14:20:53 CDT)