May 9: Final exam scores and semester grades have been posted. Thanks for giving me lots of delightful work to read in the finals.... I greatly enjoyed having you in my class. I have to keep the fional exams for one year -- but you are welcome to stop by at anytime to take a look, or just drop in to chat about math. Hope to see you again in the near future -- good luck and success, stay in touch!

Apr 28: Sorry for the missed last line of the proof -- it really was the same argument as we did in chapter 6, "without x". The main hint is that what we have (want to estimate) is (for all n, for all x) d(f(x),f_n(x)), yet we know only (at any x) d(f_m(x),f_n(x)). This immediately suggests to insert for triangle comparison the term d(f(x),f_m(x)) -- but the key is that we can use the same fixed m for all x.
[[Aside, the textbook's x_0 is what I called x, the book's F is our f.]]

Apr 18: Test 3 numerical results.

Apr 13: Job opportunity (prospective teachers).

Apr 6: Need this hint for proving lemma 9.5.4.

Apr 6: No office hours on Thu/Fri Apr 7/8 (I am leaving right after class for a conference).

Apr 6: Just a quick poll: Do you care/mind which color is used to grade your exams, tests, homework? (I have been moving away from red, but usually quickly run out of turquoise, and don't like to use blue ball pen when your work uses the same....)

Apr 5 and 6: Date for test 3 confirmed: next week Wednesday April 13

New time for office hours for BOTH
Tuesday March 29 1 and Friday April 1: 12:00-1:00.

Mar 24: Course announcement: Prep for Putnam exam etc.

Mar 23: More on writing-style. Much of the homework I am grading starts like: "suppose f is uniformly continuous. Thus for every eps>0 there is delta>0 such that ... since (a_n) is Cauchy ..." -- basically reciting the definitions, but not defining any values of the various symbols. (I.e. after this line, you still have no eps, no delta, no anything to work with!). I feel it is much more efficient -- and standard practice -- to instead of just reciting the definition, actually use the definition to define/assign variables that can be used from now on! For the above example, I prefer: "suppose eps>0. Since f is assumed to be uniformly continuous, there exists delta>0 such that for all x,y, if ....". Now both eps and delta have been fixed for the rest of the proof, but x and y are still free variables. The next step depends on what you want to prove -- it is important to understand which symbols have been qauntified (or been assigned values), and which are still free.

Mar 16: I still get far too little homework. For success in the class need much more effort... In any case, if time permits, I still would like to see a proof of the "sandwich theorem" (thm 5.1.7). The key is that the existence of the limit of the "middle" function is part of the conclusion, not part of the hypothesis. E-mail me if you would like to volunteer to present your proof.
I will collect homework for chapter 6 on Tuesday March 22, and homework for both sections 7.1 and 7.2 on Thursday March 24.
When grading the homework I still saw too much confusion when you, the prover, may choose any epsilon you wish -- and when you have to work with whatever epsilon is given to you. It is simple: When you have to prove that a limit exists, or has a certain value or the like, then for WHATEVER epsilon is GIVEN by your ADVERSARY, you have to show the existence of a delta that does something. When your hypothesis is that a limit exists, or has a certain value or the like, then YOU may choose whatever epsilon you like, and the hypothesis guarantees that there exists a delta that does something. In most proofs you will have to combine both of these as you usually assume that some limit exists, and then have to show that another one exists.
The basic structure of these arguments is very much the same for sequences, limits, contintuity, derivatives, integrals, sequences and series of functions... Learn it once, and earn points for the same basic structure over and over again.

Mar 15: I posted "mid semester grades" -- warnings for students about whose work I am concerned. As usual, I rather send too many, than too few grades. Hope that in the end, most will earn good grades....

Mar 14: Test 2 numerical results with a few comments.

Mar 4: This time I decided to grade all homework myself. First of all, I am unhappy that several students do not hand in any/enough homework. But then I did not even make it past problem 4.2/1 (so far): Just very few students got one direction reasonably right, but no work exhibited any understanding of what the other direction required. I decided to type up two sample solutions with some comments about the general logic / startegy: Limits of functions via sequences: A commented worked example.
This is hard work, it is harder than what to expect on a test -- but I expect much more effort on such homework. Many "solutions" I saw looked like 5-minute efforts --- a problem like this should reasonably take an hour at least on first encounter, and even five hours may happen. I am used to this kind of argument -- and I needed more than 15 minutes to get an argument with whose details I was reasonably happy with. (Typing it up took another hour).
On the test we will have mostly less involved logic problems, and mostly problems similar to ones you have worked before. But to earn a high grade in the class, you will have to do more. My advice: For Tuesday prepare sections 6.1 and 6.2 and be ready to present your work (assigned homework problems and exercises from text). This will be helpful as preparation for the test, too (technically there will be no Cauchy-sequences on the test, but the exercises review familiar lines of reasoning).

Mar 2: For your reference, corrected definitions and theorems. Continuity: Definitions and relation to limits/ limit points
An updated version of this hand-out was posted at 8:00 p.m. This also includes a definition of a limit point of a sequence.

Feb 24: Deleted chapter 10 from schedule and used freed up week for deeper coverage of other core topics. All homework for chapter 4 will be collected on March 3.

Feb 21: Test 1 numerical results. Some comments about test 1.

Feb 15: On Thursday I would like to see some student present a proof of theorem 3.4.9 (problem 3.4.6) before beginning to work on limit points and closed sets.

Feb 15: If for whatever reason you loose your sticker (or did not receive any), then please go to the Undergrad Math office PSA 211 to get a replacement. The test will be given in PSA 21 (basement!) on Wed 16 between 8:00 a.m and 7:55 p.m. (last starting time is 6:30 p.m.) You need to bring your SUN-card ID. There is no time limit. The test is "closed everything". For more details on testing center policies see here.

Jan 31: Some type-ohs have been corrected in the hand-out Sqrt(2) exists as a real number.

Jan 27: I have added the phone-numbers and changed e-mail addresses according to the data provided this morning. Please check the page Students for mistakes.

Jan 27: No office hours on Friday Jan 28 -- I am giving a talk at the Los Alamos Days at UA.

Jan 26: Our text is still changiung -- and type-ohs etc are inevitable... I suggest that you e-mail me any mistakes you find, and I will collect your messages and forward them to the author as a whole bundle (except very critical ones that I will send right away). But you are welcome to contact her directly, see below.

   Date: Wed, 26 Jan 2005 14:16:11 -0500
   From: Carol Schumacher 
   To: Matthias Kawski 
   
   Feel free to forward anything relevant things from me to your
   students.  They can also contact me directly, if they want.
   
   Carol

Jan 25/26:Type-oh(in textbook exercise): On page 23, Exercise 1.3.1. The third choice should be "a is a negative number" instead of "-a is a negative number."

Jan 25: I just talked to our grader (a Ph.D. math student) -- and he has one request: Please staple your homework (so pages do not get mixed up -- he grades for four classes).

Jan 21: Just for fun, some pictures of the graph of the function (x,y)-->int(sin(t)/t,t=x..y);. .mw and .mws).

Jan 19: Reserve items for our course at the library.

Jan 18: Take a look at this problem seminar.

Jan 18: For all new students who missed the first day in class. Distributed were a one-page abbreviated syllabus, questionnaire, and a diagnostic test. Please complete the questionnaire and work the diagnostic test, and turn both of them in on Thursday. The diagnostic test has some tricky questions -- try hard, but know when to stop. It is designed to help both the student and me about the level of preparation and readiness for the new material.

Jan 18: The textbook has arrived in the bookstore. We will start w/ section 1.2 on Thursday (after discussion of the diagnostic test). Please read sections 0.1 and 1.2 and start working the exercises.

Jan 17: Please send me suggestions regarding books in the library which should be placed on reserve (and for what periods).

Dec 24: The WWW-page for policies, expectations etc. and a crude week-to-week schedule of topics have been published.

Dec 21: Working on the WWW-site for the spring 2005 class.

The WWW-site for the fall 2004 class is slowly being dismantled, but bits and pieces are still alive.

Regarding the text-book, the plan is to use the newest revision of a new textbook under preparation titled "Closer and closer" by Carol Schumacher. Some students may be familiar with the book ``Chapter Zero'' by the same author, which we used in spring 2004 in two sections of MAT 300. If you have concerns or questions about the text, please do not hesitate to contact me. At this time I am still waiting for comments from students of the fall class, and also am helping with the revision of the new text.