Math 424, Fall 2016

This is the home page for Math 424, "Honors Real Analysis ". This class meets for the Fall 2016 semester at MWF 1:00-1:50 in 243 Altgeld.


The mathematical links page is here. Suggest a new one!

Class Diary, blogger style

Final Exam scores: 200(1), 190s (9), 180s(7), 170s(8), 160s(5), 130s(3).

Final Grades (I got approval for giving the number of A+'s that the class earned; that's what took an extra day): A+ (7), A(13), A-(4), B+(5), B(3), C+(1).

Do not be concerned about your "rank" in this class; I'm not sure I've ever taught a better large group than this. I was very impressed.

The Final is Wed, 12/14, from 7-10, in our usual room. Here are some questions I've answered on the Final by email

1. The Weierstrass Approximation was "bonus" material, not for the final, but you may well use it in future classes!

2. How to approach Riemann Sum questions: What you should do is set up an interval [a,b] and a function f so that the sum that is given is, up to a multiple, a sum of terms like f[x_i*] * (x_{i+1}-x_i), where x_i* is in the interval [x_i,x_{i+1}] and where x_0 = a and x_n = b. In the third test, the given sum could be rearranged into a sum of (1 + k/n)^{2/3} * (1/n), where k went from 0 to 2n-1, so this could be interpreted as f(x) = x^{2/3}, a = 1, b = 3 and x_i = 1 + i/n, for i = 0,...2n. (Or as f(x) = (1+x)^{2/3}, a = 0, b = 2.)

3. The supremum of a set doesn't have to be in the set, but it's always in the closure. For example, if S = {1 - 1/n : n = 1,2,3,....}, then 1 = sup S, because it is an upper bound, and if u < 1, then there is x in S so that u < x, so u can't be an upper bound.

4. "What is the difference between a uniformly continuous function and uniform convergence, and how that is used for power series?" These are two slightly different things. A uniformly continuous function (see p. 80) is one thing and, if f is differentiable on a set A, then f is uniformly continuous iff f' is bounded on A. On the other hand, if a sequence of continous functions (f_n) (see p.85) converges uniformly to f on a set A, then we can conclude that f is also continuous. The application to power series is that if you let f_n be the first terms of a power series, up to a_n x^n, then f_n is a polynomial (and so is continuous), and if you can prove uniform convergence on a set, then the power series itself is a continous function on that set. The other point is that if a power series has radius of convergence R, so it converges on the set |x - a| < R, and if r < R, then the convergence of (f_n) is uniform on the smaller set |x - a| \le r, and so the power series defines a continuous function on any such set.

5. " Why does the definition of continuous function involve the inverse image of an open set?" Well, that's the definition of continuous f: E -> E' for more abstract spaces. If you want to retrieve the usual definition for R, suppose f(x) = y, and let O' = (y-e, y+e), which is an open set in R if e = epsilon > 0. Under the abstract definition O =  f^{-1}(O') is open and since y is in O', x is in O. Since O is open, there is d = delta > 0 so that B(x,d) = (x-d,x+d) is in O. This means that if |z - x| < d, then z is in O, so f(z) is in O'; that is, |f(z) -y| < e. Maybe you're asking a different question. Let f(x) = x^2, which is a nice continuous function on R and let A = (-1,1), then f(A) = [0,1) is NOT an open set. This is just the way continuous functions work!

6. The Final covers the whole semester's work. In principle, there will be at least one question which might have appeared on each of the ten homeworks. Think of the list of topics for the three hour exams stapled together. In the second part of the exam, there will be 10 or 11 problems from which YOU CHOOSE 8. You can't do more and have me only count the best 8; I'll grade the first 8 I see. This is clearly covered in the instructions!

Th 12/8 -- Some review questions, lots of non-math questions as it turned out

W 12/7 -- Some review, a discussion of the Weierstrass Approximation Theorem, and a handout. This won't be on the test. ICES forms were distributed.

M 12/5 -- Test 3 returned and gone over in some detail. 3 x 5 cards for the final were distributed. Other review questions, then a variety of topics related to iteration, including the Chebvyshev polynomials and the behavior of the iterated sine, to be found in De Bruijn's book on Asymptotic Analysis.

F 12/2 -- Test 3, not online

Test 3 will be Friday, December 2, based on the material in Ch 6,7 and the handouts, and HW 8, 9, 10.

W 11/30 -- Homework 10 returned, with Homework 10 retrospective (and some bonus notes). Also talked about Test 3 and distributed key words for the test. 3 x 5 cards (for those who left quickly) are available from an envelope hanging on my door (327 AH). Second half of the class was the Cantor set and the Cantor-Lebesgue function. Differentiability is difficult.

M 11/28 -- Went over Homework 10 solutions in considerable detail and also a Bonus notes summarizing what we know about power series. What new material there was had to do with the evaluation of a power series on the boundary of its convergence, and also the fact that an absolutely convergent series can be rearranged and will always converge to the same sum.

Homework 10, due Monday, November 28; 1. A student seems confused by the role of the function g in problem 4. My reply: You might want to look at some simple examples of functions g with this property. Two such are g(x) = 2 (constant) and g(x) = 1 + x. The point is that the behavior of convergence depends of the values for large n, and for large n, 1 + 1/n is close to 1, so g(1 + 1/n) is close to 2.

F 11/18 -- Homework 9 back. Scores were high enough that no retro is needed. Went through differentiating integrals and gave some examples from Homework 8 retrospective by Prof. Kent Conrad (U.Conn). Then some examples of power series in more than one variable. Whee-ee.

W 11/16 -- I distributed Homework 8 retrospective and Bonus notes on the Stirling approximation. Otherwise I proved something about how the limsup behaves when composed with a continuous strictly increasing function and then reproduced, with variations, the book's derivation of the trigonometric functions. Hints at future work on differentiating under the integral sign and rearranging conditionally convergent series.

M 11/14 -- I distributed Homework 9 solutions and Homework 10. Class discussion involved the truth about power series, and a side trip to alternating series. I owe you some handouts, and they may show up during break.

Homework 9, due Monday, November 14. 1. Yes, the closed form in #4 may look different for smaller values of n, but it stabilizes quickly.

F 11/11 -- I distributed Bonus notes on the Cauchy Condensation Test, and a couple of examples from the 11/9 class. We also went through some of the wonderful properties of power series, and I derived the Stirling formula for n!, for which there will a handout soon.

W 11/9 -- I distributed Homework 8 solutions and talked particularly about the extra credit problem. (These solutions also include the result involving |an|^(1/n). Also a variety of techniques for rearranging the sequence 1 - 1/2 + 1/3 - 1/4 ... that will be presented in a handout later. Started on power series. Distributed Homework 9, due next Monday.


1. Yes I messed up in writing #1. "For any integer N, let P_N be the partition with x_k = a + k*(b-a)/N for k = 0,..,N."
2. In #3, when you prove that d(p,q) => p = q, that just means that p(x) = q(x) for x in [0,1]. Behavior of the functions outside [0,1] is irrelevant.
3. In this class, "log" always is the same as "ln".
4. In #2, I mean what's written. Do some simple algebraic manipulations which show that the given sum is a Riemann sum for a relatively familiar function and straightforward partition, not unlike the ones in #1.

M 11/7 -- Test 2 returned and discussed. Full speed discussion of infinite series, mostly from the book, and notes for topics that aren't covered there.

F 11/4 -- I finally distributed integration notes, and made the correction to HW8#1 noted above. Finished VI-5 and VII-1 in the book. Talked a bit about upcoming courses.

W 11/2 -- Test 2. Not online.

On Test 2
I wrote a list of key words to help you prepare for the test. (By "terms and their properties", I mean that you should know something about what these things are. You will not be expected to repeat proofs from class, especially long ones.
1. This comes from an email. How to think about lim a_n, limsup a_n and liminf a_n: Suppose lim a_n = A. This means that for any e > 0, a_n will eventually stay between A-e and A+e, Suppose liminf a_n = B and limsup a_n = C, B < C. This means that a_n will be buzzing between B and C. Eventually it will stay between B-e and C+e, but infinitely often it will be less than B+e and infinitely often it will be more than C - e.

M 10/31 -- I distributed and talked about the list and did a couple of review examples. Nothing came up to change the list that was handed out. At the end, I talked about the log and exponential functions a bit, a topic I'll return to on F 11/4.

M 10/24, W 10/26, F 10/28 -- We spent most of the week talking about integration, and how the definition from the notes distributed on 10/19 (and mandated by the syllabus) can be reconciled with the definition in the book. It turns out that they are equivalent, and there will eventually be notes on the painful steps required to take it.
Here are three of the handouts (updated 10/31 with HW 6 retro) there wasn't enough to say about the HW 7 retro to write it up): Homework 7 solutions, Homework 8 [actually, I forgot to distribute this!] Homework 6 retro.

Homework 7, due Monday, October 24.
1. If you use a theorem from class or the book correctly, you do not have to prove it again.
2. In problem 2, I want you to find *specific number* r, *using the information provided*, so that a non-trivial identity of the form sup|f'| \ge r or \sup |f'| \le r is correct. Yes, saying that sup|f'| \ge -7 (say) is both true and uninteresting and won't receive full credit!

F 10/21 -- I distributed the HW 5 retrospective and talked more about integration. We're close to the end of the notes which were distributed on 10/19.

W 10/19 -- Homework 5 was returned; retrospective on Friday. We started to talk about integration, using Riemann-Darboux discussion from another book . I will talk more about integration, and tie it in with the text, on Friday. This lecture started the material which will be tested on the third exam.

M 10/17 -- Homework 6 solutions were distributed, as was Homework 7, due Monday, Oct. 24. We went through the proof of Taylor's Theorem, and gave a few illustrations. Brief discussion of the Cantor set and the Cantor function, and yes there will be notes. The material on Homework 7 is the last of that which will be fair game for the second exam, probably coming in early November.

Homework 6, due Monday, October 17.
1. Be sure to look at the comments on the 10/14 handout.

F 10/14 -- Started with a set of bonus notes which (a) made a few corrections and interpretations to the homework (b) gave the full proof for HW5#2 and (c) gave an alternate proof of the Chain Rule. Also some applications of the Chain Rule. The Taylor Theorem awaits!

W 10/12 -- The first exam was discussed and returned, and lots of further talking about differentiation.

M 10/10 -- Homework 5 solutions were distributed, as was Homework 6, due Monday, Oct. 17. Test 1 not returned; aiming for W 10/12. Material covered was on differentiation. The very next topic we will cover is the chain rule, soon to be followed by inverse functions (briefly) and then onto Taylor's Theorem.

Homework 5, due Monday, October 10. Comments will be pinned here.
1. Yes, I forgot something important on #1: The last sentence should be: If (E,d) is a metric space with an isolated point p, prove that it is not connected.
2. You should probably use in #2 that b_n > 0 for sufficiently large n.

F 10/7 -- We moved on and finished chapter 4 (skipping msot of the discussion from the bottom of p.87 to p.90 on the metric space of continuous functions on a compact metric space.

W 10/5 -- The first exam. Never put online!

First exam, Wednesday, October 5, in class. 1. Bring questions to class on Monday.
2. On Monday, I wrote a list of key words to help you prepare for the test. (By "terms and their properties", I mean that you should know something about what these things are. You will not be expected to repeat proofs from class, especially long ones.
3. You are allowed to bring one (1) 3 x 5 card, and you can write on both sides if you like, but no books, no notes and no electronic devices.

Homework 5, due Monday, October 10. Comments will be pinned here.
1. Yes, I forgot something important on #1: The last sentence should be: If (E,d) is a metric space with an isolated point p, prove that it is not connected.

M 10/3 -- Gave a list of key words to help you prepare for the test, and did some review, answering a bunch of questions. New material was section 4.5 and a hint at 4.6. Nothing on continuous functions will be on test 1.

F 9/30 -- Returned Homework 4 with retrospective and bonus notes and responded to some questions. The new material covered was chapter 4, section 4, involving uniform continuity. I'll be happy to answer all questions on Monday, and if you run out of them, we'll move on in chapter 4.

W 9/28 -- A bonus notes covering a couple of items from 9/26 class. We zipped through chapter 4, sections 2 and 3. At the end, Homework 5, due on October 10.

M 9/26 -- It was decided that the exam would be W 10/5, in class, 50 minutes. Homework 4 solutions were distributed, and some things amplified upon. We went through Chapter 4, section 1. Next up: sections 2 nad 3. Nothing from Chapter 4 will be on the first exam.

Comments on HW 4, corrected, pinned to the top until it's due.
1. In problem #2, R is really meant to be E^1.
2. In problem #2b, K_n is a subset of R, not an element, and you want to show that K is NOT compact.

F 9/23 -- I distributed a lengthy bonus notes, which covered the topological discussions of 9/21 and gave information about the limsup and liminf, which can otherwise be found in the the text (III,HW18,19). Homework 3 back, with retrospective. At the end of the class, we began Chapter IV, on continuous functions.

W 9/21 -- I distributed a corrected Homework 4. Please delete previous one. Class involved more on connectedness, maximal open subsets and an introduction to lim sup and lim inf. Massive handouts for 9/23.

M 9/19 -- Homework 3 solutions were distributed, as was Homework 4, due Monday, Sept. 26. [Note: corrected on 9/21.] Also, bonus notes covering the classroom proof that a bounded and closed set in R^2 is compact. In class, we finished what we are going to say about compactness and talked about connectedness.

Comments on Homework 3 FINAL VERSION
1. Yes, 4b is intended to be an if and only if statement. As mentioned in class, there is a familiar geometric fact you can use if you like which you quote without having to prove.
2. For #6, I'd like an explanation, along with enough of the construction to show the pattern. No proof longer than 1 paragraph.
3. For #1, several email questions. I won't give specific hints or advice, but will remark that problem #1 is supposed to be the easiest and there is a fairly direct short proof using material from class.
4. For #2, a rearrangement is not a subsequence. Use the definition of convergence, knowing that (an) converges to prove that (bn) converges.

F 9/16 -- Still more on compactness, with a different and hopefully more intuitive proof that closed bounded S in the Eulidean plane is compact. (Handout later.) Also, Homework 2 back, with retrospective and bonus notes covering the earlier class proof that [0,1] is compact.

W 9/14 -- More on compactness, mostly following 3.5, and another handout, mostly of corrections, but also a proof of a useful result for problem-solving which is not in the book.

M 9/12 -- Homework 2 solutions were distributed, as was Homework 3, due Monday, Sept. 19. The first exam time will not be discussed until we've finished covering the material. In class, we proved that R (and R^n with the Euclidean metric) are complete metric spaces, and started talking about compactness. More on that W.

Comments on Homework 2 FINAL version
1. Someone wanted a clarification of #3. Here is a restatement with fewer symbols:
The problem says that if p1 and p2 are two different points in a metric space (X,d), then there exist open sets O1 and O2 so that p1 is in O1, p2 is in O2 and the *intersection* of O1 and O2 is the empty set; that is, O1 and O2 have no points in common.
2. On #2, we do not know what a continuous function is yet, so in particular, we do not know that the square root function is continuous.
3. On #5, one can answer d without using a, b or c, although one should use the definition of the sequence (!). There are actually many ways to view this problem, from various courses. Also, correct non-induction proofs will be accepted. (Make sure they are fully explained.) What I mean by a closed formula is something where you can plug in n and get the value of the sequence directly, and not by using the recurrence. You have to prove it to get full credit! EG (not correct as an answer) a_n = (e^n + (-17)^n*n^2) / (Cos[n] + n^2).

F 9/9 Homework 1 back, with retrospective. Bonus notes on topology, as promised. The class itself reviewed these and talked about the convergence of monotone sequences in R and the existence of Cauchy sequences in general metric spaces.

W 9/7 -- Return from break. Homework will be returned on Friday. Most of the class discussed topology, with notes that will be passed out on Friday. We are in sections 3.3, 3.4.kosdp[af09g0eru89

F 9/2 -- Homework 1 solutions were distributed, along with the hard copy of the email list. Your instructor left his notes in his office but still managed to be coherent for most of the hour, covering pp.44-50. Read the end of 3.3, and then 3.4 and 3.5 for Wednesday. A free-association discussion of the metric space of error correcting codes at the end. Notes maybe later. In any case, Homework 2, due Monday, Sept. 12, was distributed. Have a good break.

Th 9/1 -- Homework comments (final list, a short one)
#3 -- Yes, the triangle inequality is your friend.
#5 -- Just to be clear: these d_1 and d_2 are not the specific distances for R^n. If it makes it clearer, call the metric spaces (X,d) and (X,d').
EC -- No hints, but there are a number of different approaches. I'll put all of them in the solution sheet tomorrow.

W 8/31 -- Not too many questions on the homework. Finished section 3.2 on open balls and open sets and started on convergence. Will update with homework commentary, if any, on Thursday afternoon.

M 8/29 -- Sign-up contact sheets were distributed after class by email. (Won't be online.) The first set of bonus notes was distributed. We have now covered through section 3.1. On Wednesday, we'll talk about section 3.2 and maybe 3.3. Bring questions.

F 8/26 -- Decimal representation, the existence of square roots in R (but not necessarily in Q) and Cauchy-Schwarz. We'll start Section 3 and metric spaces on Monday. Math 424 HW1, due Friday, Sept. 2, was distributed. Rules for homework are on the course organization. If you prepare your paper electronically, please sign your name. If you can't make it to class when the hw is due, you can get it to me earlier, or under my office door, or in my mailbox or as a .pdf email, but please scan with higher resolution than a random cellphone.

W 8/24 -- We zipped through chapter two up through 2.3, and talked about the existence of least upper bounds. Outline maps of various places were passed around and people marked their home towns.

M 8/22-- First day of class. The questionnaire was distributed. We agreed on Monday as a due date for homework, but not for the first couple of weeks. Handouts: Course Organization and How to Solve It guide, as well as an emergency guide, which I got from the UIPD. Read Sections 1.1, 1.2, 1.3, 1.4, 2.1, 2.2 of Rosenlicht for Wednesday and bring questions for class.