Math 424 F Home Page

This will be the home page for Math 424 F -- "Honors Real Analysis" This class meets for the Fall 2019 semester at MWF 2:00-2:50 in 345 Altgeld.
The mathematical links page will be here. Suggest a new one!

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

F 1/3 -- The link below doesn't work any more. Send me email if you want a copy. This webpage itself will disappear before the end of January 2020.

W 12/25 -- Here are the Incomplete Solutions to the final, along with a (corrected) copy of the test. I will be taking this page offline on 1/2/20, so after then, please just write me to get a copy.

F 12/20 -- Report. You were an excellent class! On the Final, the distribution was: 200 or more (3), 190s (5), 180s (6), 170s (7), 160s (4), 150s (2), less than 150 (1). The course grades were A+ (6), A (11), A- (4), B (6), B- (1). Once I finish grading myT other class, I will write an incomplete guide to the final. The most popular questions to skip were #12, #13 and #10 (in that order), but those weren't the hardest problems! Have a great break! If you want to learn your specific information, please email me with your booklet number as PIN.

M 12/16 -- Final exam.

W 12/11 -- Final thoughts and the ICES forms.

M 12/9 -- Test 3 and HW 10 returned. Some discussion on series and other related topics, and a very quick description of the Weierstrass Approximation Theorem, and its generalization, the Muntz-Szasz Theorem; see the wikipedia article.

F 12/6 -- Test 3. Not online.

W 12/4 -- I talked about major topics for the third test, on Friday. Passed out old notes on the Stirling approximation to n! Also talked a bit about the Stieltjes integral, which won't be on the test either.

M 12/2 -- The Tenth Homework Solutions were distributed and a few problems were discussed. 3 x 5 cards were distributed. Most of the class period was a quick introduction to general measure and Lebesgue integration. This was taken from H. L. Royden's "Real Analysis"; there are many other good books, including the ones by Walter Rudin. Test 3 will be F 12/6.

Homework 10 clarification

  • On problem 2, s_{n^2} = a_1 + . . . + a_{n^2}. (That is, I'm taking the sum of the first n^2 terms.)

    F 11/22 -- A freeflowing discussion of a lot of different topics, most related to mathematics, research and teaching, rather than 424 specifically.

    W 11/20 -- Note Homework 10, below. I talked about the Cantor function and distributed a brief I also talked about the asymptotics of the function f(x) = sin x and the remarkable theorem, due to De Bruijn that if: a(0) is in (0, pi) and a(n+1) = sin(a(n)), then
    a(n)/( (3/n)^(1/2) ) --> 1 for any value of a(0).

    Homework 10 is here.

    M 11/18 -- The Ninth Homework Solutions were distributed. Homework 10 will be out by 11/20. Most of the class was taken up with describing representations base b, and then applying it to the Cantor function. (There will be notes later, but it's actually in Wikipedia!)

    F 11/15 -- Various odds and ends. HW 7 finally returned (my apologies). Fun with summations and the growth of H_n. Series like the cos and sin, but with every m-th term taken, not just for m=1 or m=2.

    W 11/13 -- Ninth homework distributed (see below). More on power series plus the Cauchy condensation test and the fact that all derivatives of e^(-1/x^2) go to 0 as x->0. We'll be finishing up chapter seven in the next few days.

    T 11/12 -- Here is the revised Ninth Homework; problem 1 is different.

    M 11/11 -- The Eighth Homework Solutions were distributed. Lots more on limsup, why n^(1/n) --> 1 as n goes to infinity and other good stuff. HW 9 will be out on 11/12, and you are urged to write me with suggestions for material to go on handouts.

    F 11/8 -- More on series and power series.

    W 11/6 -- Test 2 returned. Discussion of 7.1 and 7.2. The limsup and liminf make their triumphant returns.

    M 11/4 -- Series and convergence tests; a mixture of 7.1 and 7.2. An unexpected excursion to delta functions at the end. The Eighth Homework was distributed, due M 11/1.

    F 11/1 -- Test 2. Not online.

    Test questions.

  • A student asks for intuition about uniform convergence of functions and what happens when it doesn't occur. My reply: "The intuition is that, if f_n —> f on a set A, but the convergence is not uniform, then sup{ | f_n(x) - f(x) | : x in A} is not going to zero as n goes to infinity. The canonical example is f_n(x) = x^n on [0,1]; f(x) = 0 on [0,1) (f(1) = 1), and f_n(1/2^(1/n)) = 1/2. The theorem is that uniform convergence of continuous functions to a limit implies that the limit is continuous."

    W 10/30 -- I talked about major topics for the second test, on Friday. I did some review, described a dubious job offer to teach math in China over the summer, and completed the proof of the Stirling approximation. (Handout to follow.) I talked about an interesting property on not-absolutely convergent series whose terms go to zero. More later.

    M 10/28 -- The Seventh Homework Solutions were distributed, and much of the class was devoted to going over them. We started on infinite series, and will continue after the test.

    On HW 7

  • On #4, to give you an idea of what's going on, what happens if f(x) = x^2?

    F 10/25 -- More concrete implications. The definition of the log and the exponential, and an application (assuming HW7 #5 to be true) leading to the Stirling formula.

    W 10/23 -- Transition from the notes material to the book. The book has two definitions of integrability, which it shows are equivalent, and I made them equivalent to the book's definition. Having done that, on to the Fundamental Theorem of Calculus and implications.

    M 10/21 -- The second hour exam will be on F 11/1. Same pattern as the first, but no scratch paper except the back of the sheets. The Seventh Homework and the Sixth Homework Solutions were distributed, and I went over a few of the hw problems. We are basically finished with the handout, and will reconcile it with the book on Wed.

    -- HW 6 Comments

  • On problem 3: I would recommend trying to estimate the expression. You might get two estimates, depending on whether h is rational or not. See if you can combine them. This is better than combining the separate limits, because we haven't defined lim_{h->0} where we restrict to rational h or irrational h.
  • Also, the questions are different. You proved g continuous at x=0 on the last homework. On this one, I want differentiable at x =0 and continuous at x=2. You might be able to guess about what's on the next homework!
  • For #5, it is helpful to draw a picture and some examples of functions with the desired conditions. You have to put several things together, but it's not too hard.
  • For problem 4, we don't have the Inverse Function Theorem in the class, but difference quotients or Taylor's theorem seems like a good idea. The problem with the Chain Rule is that we don’t know automatically that the inverse is differentiable.

    F 10/18 -- I corrected the errors from 10/16, gave an alternate proof of the chain rule, and got mostly done with the notes. After we finish, we'll try to reconcile these with the definition in the textbook.

    W 10/16 -- An interesting day, because half an hour before class, I realized that, according to the syllabus, I should be doing a different definition of the integral. Here are some notes from D'Angelo and West. As might be expected, I had some gaps in my presentation, fixed on F.

    M 10/14 -- The Sixth Homework and the Fifth Homework Solutions were distributed, and I went over a few of the hw problems. Mathematically, we are now into the definition of the Riemann Integral, and a few of its basic properties.

    -- HW 5 comments:

  • On problem 6, an alert student observes that if a_n is an increasing and bounded sequence then lim a_n = L exists, and L will have to be a cluster point too. So either assume this, or assume that a_n -> infinity.
  • Also, the choice of "10" is not completely accidental, in my example at least.

    F 10/11 -- Two links. The first is to a video on patterns of putting points in polar coordinates that naturally brings up a range of mathematical topics. The second is to the land acknowledgement, as presented by the Chancellor at the start of every Senate year, and inspired here by the upcoming "Columbus Day".

    F 10/11 -- The first test was returned and discussed. We finished differentiation, with a discussion of Taylor's theorem and intimations of how it will be useful in power series.

    W 10/9 -- HW 4 returned. We got through a lot on differentiation, up to the Mean Value Theorem.

    M 10/7 -- Back to the course. I expect to return HW 4 on Wed. and the test on Fri. I brought the Fifth Homework, but forgot to distribute it until after the bell. We finished chapter 4 (skipping the last few pages), and started on differentiation.

    S 10/6 -- Here is a link to the last spring's course description of Math 496. I expect to teach it in the spring, but nothing is definite yet.

    F 10/4 -- Test 1. Not online.

    W 10/2 -- Ideas and theorems for the first test, on Friday, were discussed and various problems were worked through. No new material was presented.

    M 9/30 -- Homework 4 collected and the Fourth Homework Solutions were distributed. I continued to talk about uniform continuity and started to talk about uniform convergence. The test is on F 10/4. On W 10/2, I will give a list of key terms and properties to help orient your studying. You are allowed to bring one (1) 3 x 5 card (distributed on W), and you can write on both sides if you like, but no books, no notes and no electronic devices.

    F 9/27 -- Introduced the idea of the "consent agenda" for fairly intuitive things, like the rules on combining limits and got part of the way through section 4.4: continuous functions on connected sets, and the introduction of the idea of uniform convergence. First test is F 10/4, so HW 5 will be delayed, but HW 4 is still due on M 9/30.

    W 9/25 -- Redid the proof that f is continuous on E if and only if the inverse image of open sets is open and worked through section 4.2 and sequential convergence.

    HW 4 questions

  • An alert student observes that I forgot a hypothesis in #3. Assume that (E,d) has at least one isolated point. To make it easier on the grader, say that p_0 is an isolated point.

    M 9/23 -- Finished with connectedness and starting with continuity. The test will run through the end of Ch.3. I forgot to distribute HW4 except to people hanging around at the end, and will bring it on Wednesday, but here are the Fourth Homework and the Third Homework Solutions.

    HW 3 questions

  • A student asks if proofs are needed in #5. My reply: "If I ask you to give an example of an object with property P, you should include a proof that it, in fact, has property P. It doesn't have to be very detailed."
  • I have gotten a number of questions about #1, asking for a hint. This is a summary of what I said: "It might help for you to think about how this problem could relate to the course and to the material we've been covering this week. It's definitely an "aha", and I don't want to take this opportunity away from you. The proof isn't very long."

    F 9/20 -- More on compactness and connectedness. Old notes with a proof that bounded and closed imply compact in R^2. They generalize easily to R^n. The book is not very clear on the definition of connectedness, because "open" here means open in the set you are checking for compactness; this is sometimes called "relatively open".

    W 9/18 -- HW 1 returned. Some "practical" results on limits: a_n -> 0 ==> | a_n | -> 0; if |r| < 1, then r^n ->0. If |a_{n+1}-L|/|a_n-L| \le r < 1, then a_n -> L and if |a_{n+2}-a_{n+1}|/|a_{n+1}-a_n| \le r < 1, then {a_n} is Cauchy and so has a limit. If you want a .tex file of the homework, send me an email. I don't want to post it. I gave a direct proof that [0,1] is compact, and distributed an old handout with the proof.

    Tu 9/17 -- Here is the Third Homework. Paper copies will be distributed in class tomorrow.

    M 9/16 -- The Second Homework Solutions were distributed. In addition, I presented at the board the solution to the ungraded problems and talked more about compact sets in metric spaces. On Wednesday, I will give a special case of Heini-Borel: proving that [0,1] is a compact set, before doing the full proof. Also, HW1 should be returned on W.

    Homework 2 comments:

  • On problem 5, we should have a0 = 4 (not 2). My apologies for the typo.
  • Also, recall that the absolute value appears in the definition of limit and | |x| | = |x|
  • (-1)^n is often useful in describing x_n when the behavior is different for even n and odd n.

    F 9/13 -- Completion of the proof that R is complete and the extension to R^n. A beginning of the discussion of compactness, which is a strange definition. I'm aiming to get HW1 back on Monday; HW2 is due (see comments above) and HW3 will be distributed. The phone/email list (not linked) was distributed.

    W 9/11 -- Monotone sequences, complete spaces, and the reals are complete. The proof I used involved limsup and liminf which are barely in the book, and in a weird formulation. Once I write some notes on them, I'll talk about them some more.

    M 9/9 -- The First Homework Solutions were distributed. The Second Homework was also passed out. In terms of new material, we are progressing through the definition of convergence and some of its properties. Next stop: monotone sequences.

    F 9/6 -- Much more topology, up to the end of section 3.2. I talked about comparable metrics, and when I write up notes with the 1-norm and 2-norm and infinity-norm, I'll put that definition in as well; for now, it means that if d and d' are two metrics on the same set X, and there exist positive r and s so that r*d(x,y) \le d'(x,y) \le s*d(x,y), then d and d' are called comparable and the theorem is that they determine the same topology: a set is open in (X,d) if and only if it is open in (X,d'). Monday: HW 1 due, HW1 solutions distributed, HW2 put out and the phone number / email list will be circulated one last time.

    HW 1 comments:

  • On #5, d_1 and d_2 are just two distances, not necessarily the "L_1" and "L_2" ones we were discussing in class.
  • I intended in #5 that r > 0!

    W 9/4 -- More examples of metric spaces: to wit, R^n with the "1-norm" and "infinity norm", as well as the metric space of n-tuples from an alphabet, where the distance of two n-tuples is the number of places in which they differ. The relatively familiar definitions of an open set in a metric space. More examples on Friday. If you want notes on any of the unfamiliar material, please let me know.

    M 9/2 -- Labor Day, so no class, but here is the First Homework, which is due 9/9. I will distribute these in class on W 9/4.

    F 8/30 -- I started with a "close reading" of the book's proof of the existence of square roots, with a handout (not given here) of two pages of the book, and then an exposition of what was missing. Moving to chapter three, we defined metric spaces and gave some examples. The real challenge was proving Cauchy-Schwarz, using what "quadratic polynomial approach" (found in the book), and also the "sum of squares" approach.

    W 8/28 -- Observation: many questions you were asking me after class could have been asked in class. I'd prefer that. I WANT you to interrupt if what I am saying is incorrect or unclear. For example, At one point, I wrote "-N < |x| < N" when I should have written "-N < x < N". Tell me this. Class should be interactive. I don't want to be replaced by a videotape!
    Homework 1 is postponed to be due M 9/9, because there won't be enough covered material for an assignment otherwise. We talked a bit more about the properties of |x| and then moved on the the LUB axiom: every set in R that is bounded above has a least upper bound. We then took implications of this (all in the book).

    M 8/26 -- Introductions. Handouts are: Course Organization, How to Solve It guide, Class questionnaire. and information about the links page (see above.) We covered, basically, pp.1-22 of the text. Please ask if you have questions. I did some "enrichment" material on ordered fields, which won't be on the exams. We agreed to have homework due on Mondays, though with Labor Day next week, HW1 will be due W 9/4 (and available on 8/28) [see correction above]. For more on ordered fields see my short expository article: What does `>' really mean?