# Teaching Introductory Real Analysis

I am currently helping teach an introduction to real analysis course at UC Berkeley. The textbook we are using in Rudin's "Principles of Mathematical Analysis" (aka baby rudin).

I am trying to find ways to help the students understand the material better. My jobs include

1. Writing solutions to the homework exercises

2. Finding other examples that can supplement Rudin, as Rudin sometimes doesn't present enough examples

3. Suggest Problems for the Midterm and Final Examinations

For (2), I have found Kenneth Ross's book (Elementary Analysis: The Theory of Calculus) very helpful. Does anyone suggest any other books, that do a good job in "holding your hand and walking through" with various concepts in analysis?

Also, this is my first time teaching, so I am wondering how a good solution set should look like. Should it just show the formal solutions, or should it also tell the students a little bit about how to approach the problem and build intution. Or is that too much writing that will distract and be unmotivating for the student to read through?

Also, each week we assign about 10-14 problems, but only 3 problems are graded. My solutions are to the 3 graded problems. I believe there is some advantage to this system, but do you I am putting the students at a strong disadvantage if I do not write all of the solutions? Time is limited sometimes, but I plan on writing the solutions to some other problems other than the graded problems. So:

How do you decide which problems are worthy of writing solutions to (assuming time is limited and I don't have time to write solutions for all 14 problems)?

Does anyone have good ideas to supplement Rudin(which I think is a great book, but my students may disagree) to the 40 student undergraduate class? This is a sort of a broad question. I'm wondering what others did in their previous experiences, if any, while teaching a class with that book?

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Disadvantage with respect to whom? That's not the issue. If a problem involves something that could be thought of as a new idea, a solution should be made available. Or more than one solution. Sorry! –  André Nicolas Jul 9 '11 at 3:02
"If a problem involves something that could be thought of as a new idea" - can you be more specific? A lot of the problems just put together different theorems to solve a problem. –  r.g. Jul 9 '11 at 3:11
But that's not the hard part. The really unpleasant, time-consuming, and indispensable part is providing individual comments on students' approach and style. –  André Nicolas Jul 9 '11 at 3:11
As a teacher I've always thought that write-ups that discuss how to approach a problem and think about the concepts involved are far more useful than those that merely give a formal solution. I also think that it's good to present multiple reasonable approaches when they exist. And I'd certainly write up as many solutions as I could manage, giving priority to the least straightforward. –  Brian M. Scott Jul 9 '11 at 3:23
I second the recommendation of Pugh's book given in an answer below. This was the book I used last summer when teaching Math 104, and I believe students found it to be very well written. This is also an excellent source of exercises. As for extra examples, may I suggest the internet :) For instance, some of the material on my website (math.berkeley.edu/~scanez/courses/math104) may be useful. Good luck! –  Santiago Canez Jul 9 '11 at 5:39

Both Pete-if you mind me calling you 'Pete',Dr.Clark,I apologize-and Jesse have given you terrific advice. I'd like to chime in and add my 2 cents on the matter.

Firstly,I also have Körner's book and I like it immensely.The book's frequently been attacked,strangely enough,for not being very original or creative in its choice of topics and organization. I do not understand this critique.It's a book for an intermediate undergraduate real analysis course-how creative could it be without defeating its purpose?!? It's a great supplement to little Rudin and asks a lot of the deep questions Rudin avoids. I also agree Gelbaum and Olmstead is a necessity on your desk for such a course and now that it's in Dover,there's no good reason not to have it.

Next,I'd like to recommend an alternative to Rudin. I know,the purists in the audience want to lynch me right now. Unfortunately,the sad fact of the matter is that most students today-especially in the United States-just aren't what they used to be in terms of discipline or background training. But my objection is deeper then that. Even if you're fortunate enough to have strong students,they really won't get as much out of the course using Rudin simply because it's so pristine and terse. They'll be able to do all the problems,understand the definitions and theorems. But I'll bet most of them walk out of there not understanding *why*it's necessary to build all this machinery into calculus.I saw a review of little Rudin at Amazon with some student praising the book as The New Testament of analysis and one of his comments gave me pause:" Don't let people steer you away from it with motivation. Motivation is for those too feeble-minded to handle real math. He's not selling you a car,Rudin-just do the book and the you will learn The True Way of Math." Or something to that effect. I think this attitude shows exactly what's wrong with using the book even with strong students.

The book I'm recommending was written-ironically-by Charles Chapman Pugh,who taught the honors analysis course at Berkeley for over 30 years. It's called Real Mathematical Analysis and I affectionately refer to it as Rudin Done Right. The book is pitched at the same level as Rudin, with exercises just as difficult if not more so. So what's different about it? Firstly,Pugh writes in a beautiful,conversational yet very concise style that's immensely personable,contains many references to the mathematical literature and yet is amazingly curt and direct. There's very little chatter.So how is Pugh able to be concise like Rudin and still be very informative and illuminating? Pugh has a remarkable gift as a textbook author:He seems to know instinctively exactly how many words it takes to explain something,not one word more, not one word less. For example,when defining the Cantor set,he refers to it as "Cantor dust". This phrase is extremely insightful in picturing the Cantor set for the first time.This intentional parsimony,where he carefully chooses,weighs and measures every word and phase,is one way he does it. The other is the book is full of pictures-there's literally a picture on every other page. But what's amazing here is that none of the pictures are throwaways or space fillers-every single picture is very specifically designed to make a point. For example,when proving that in a general metric space,every open ball is itself an open set,he presents it alongside a picture of an open disk in the plane and how a smaller open ball can be constructed anywhere inside an open ball. The book is loaded with beautiful pictures like that and if Rudin had done half of this in his presentation,the book would have been so much more approachable. It also covers certain topics vastly better then Rudin-his chapters on function spaces and their applications to differential equations,theoretical multivariable calculus and the Lebesgue are infinitely better then Rudin's.

Run right out and order this book for your students.You'll thank me later.Trust me.

Anyway-that's my very lengthy 2 cents on the matter. Good luck,let us know how it went!

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No, Andrew, I don't mind at all if you call me "Pete". This is both the default convention of the site and my own personal preference: neither I nor the people I interact with professionally are likely to forget that I have a PhD. (By the way, I deduced your identity immediately from the lack of spacing after commas. I checked your profile and saw that you're not trying to be anonymous here, but if ever you are, you have some identifying traits to shed...) –  Pete L. Clark Jul 9 '11 at 5:19
@Pete As I'm sure you can tell from my style and context,I wasn't really trying to hide my identity in this case.But there are times I'll want to and you're absolutely right because of it. I'll work on it,thanks.: ) –  Mathemagician1234 Jul 9 '11 at 6:01
@Pete L. Clark: It also helps that he has a website that says who he is. –  Damien Jul 9 '11 at 6:37
@Mathemagician1234 I disagree that "motivation" is necessary when it comes to studying mathematics. I think that if one puts enough time and effort into reading Rudin (for example), one will be able to discover the motivation on one's own. Finally, if one is patient, then the motivation will definitely come. For example, while reading Rudin's Real and Complex Analysis, I accepted to learn measure theory in the first 8 chapters and I was amply rewarded when I studied harmonic function theory in chapter 11; the theory is perhaps the most exciting I have seen in analysis ... –  Amitesh Datta Jul 9 '11 at 9:08
@Amitesh (continued) We can provide motivation without "watering down" the material. One way is to do what Pugh does-provide lots of pictures that work completely "in synch" with the concise text.Another way that I'm a firm believer in is historical context. The most informative classes I've ever been a part of-both as a chemistry and a mathematics student-were those in which the story of the discoverers of the facts was told along with the facts.Not only is the story incredibly compelling in and of itself,it shows what questions were asked that lead to their discovery. –  Mathemagician1234 Jul 16 '11 at 5:31

De​ar Ro​han,

First let me say that I got a little jealous after reading your first paragraph! What you are about to do should be very rewarding and a lot of fun.

I think you are right to be concerned that many students will find Rudin's book [as an aside, I don't really like this "Baby X" stuff: it seems not so subtly discouraging to describe university-level texts in this way; you could say either Rudin's Principles or, if you must use an epithet, "Little Rudin", because it is indeed a smaller book than his other analysis texts] too terse in and of itself. Sufficiently strong students will consider it a rite of passage and adapt to it eventually, but the entire current generation of "off-Rudin" undergraduate analysis texts seems to be fairly convincing evidence that the average undergraduate needs somewhat more help. Which is not to say that Berkeley is populated by average undergraduates, but I think even very strong students, whether they realize it or not, could learn more efficiently if the text is supplemented. (It happens that this was the primary text used in the math class I took during my very first quarter at the University of Chicago. It wasn't completely impenetrable or anything like that -- much less so than some of the lectures later on in the course! -- but I think I would have benefited from some of the supplementation you describe. In any case I'm too old to give a really independent evaluation of that book now: I've had it for getting on 20 years and have read much of it backwards and forwards countless times.)

The good news is that the off-Rudin phenomenon is so widespread that there are almost infinitely many places to go for a source of more problems, examples and so on. You really can have your pick of the litter. But since you asked, here are two books I like a lot, one old and one rather new:

Gelbaum and Olmstead, Theorems and Counterexamples in Mathematics.

What it claims is what you get, and what you get is very valuable. Asking students for counterexamples is a great way to keep them alive and awake in such a course: it's so easy for a young student to get snowed under by the barrage of the theorems and not to appreciate that so many theorems in real analysis have somewhat complicated statements because the simpler statement you are hoping for at first is simply not true. Coming up with counterexamples really helps students participate in the development of the material: if you don't do any of this explicitly, the very best students will do some of it on their own, but for a lot of the students learning the theorems will amount to a lot of arduous memorization.

Körner, A Companion to Analysis....

This is a pretty fantastic book: a long, chatty text much of which fills in nooks and crannies and works very hard to get the student to appreciate why things are set up they way they are. For instance, by now a lot of instructors have realized the pedagogical need to provide more up-front motivation for the real numbers and the obviously important but initially mysterious least upper bound axiom. Körner's book carries this line of thought through more deftly and thoroughly than any other I have seen. He asks the question "What happens if we try to do calculus on the rational numbers?" and he comes back again and again to answer it. It's tempting to throw out an example of a continuous function on a closed interval which attains its maximum value only at an irrational point and just move on, but this leaves a lot of cognitive work to the students to really appreciate what's going on. Körner does much better than this. Moreover, Körner's book ends with the best list of analysis problems I have ever seen. There are literally hundreds of pages of problems, thoughtfully organized and appealingly presented. This is an invaluable resource for someone trying to flesh out an analysis course.

I see that I've now written at length and not addressed most of your questions, which concern the solution sets. That may be for the best: it's been a long time since I regularly wrote up solutions to problem sets. This takes me back to my undergraduate days as well, when they were handwritten and mimeographed: yes, that was a pretty strange thing to do even back in the 1990's. (I will advise you tex up the solutions rather than handwriting them, although even this is not as de rigueur as one might think: I have colleagues who think that handwritten solutions are more appealing. I think they're crazy, but oh well.) I didn't get any feedback on them and often wondered if they were actually being read. So I had better leave this for someone else to advise you.

Good luck!

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+1 An excellent and insightful answer! –  Amitesh Datta Jul 9 '11 at 3:50

Before I begin, a disclaimer: I am an undergraduate who has never officially taught an upper-division math course before, so perhaps you should take these suggestions (and opinions) with a grain of salt. (But for what it's worth, I do think I spend an inordinate amount of time thinking about pedagogy and curriculum design, especially as pertains to undergraduate analysis classes.)

Suggestions:

(1) Examples. Saying that Rudin "sometimes doesn't present enough examples" is, I think, a bit of an understatement. Personally, my inclination is always to present an example (and non-example) of a definition as soon as it is introduced.

For examples regarding metric space concepts, you should definitely check out Metric Spaces by Searcoid. This book follows a somewhat unconventional approach to the topic, but is loaded with examples.

Another great book is Counterexamples in Analysis, whose utility I cannot overstate.

(2) Provide motivation. In a first course on real analysis, it is not unusual to see students wondering what exactly the point of it all is. Rudin also does not provide much in the way of motivation.

I often like to explain that at least one of the goals is to provide a rigorous treatment of calculus. Providing counter-examples is also a good way of illustrating how all of these seemingly pedantic details are in fact quite necessary. On a related note:

(3) Name the concepts. For example, in Chapter 2, Rudin proves that compactness implies limit point compactness... except that the term "limit point compactness" is never actually used. As a result, the theorem (2.37) seems rather arbitrary; why should we care? Giving concepts names can help emphasize their importance.

I myself first learned real analysis from Baby Rudin's Principles. Here are some things that I really wished Rudin had mentioned:

• Fact: If for all $\epsilon > 0$, we have $a < b + \epsilon$, then $a \leq b$ -- assuming that $a$ and $b$ are independent of $\epsilon$.

This fact is implicitly assumed in countless $\delta$-$\epsilon$ proofs. As a student, I found it mystifying why we could sometimes "get rid of the $\epsilon$" and sometimes we couldn't. This fact explains it.

• Fact: $a = \sup S$ iff: a) $a \geq x$ for all $x \in S$, and b) $\forall \epsilon > 0$, $\exists x\in S$ such that $a < x + \epsilon$.

This one fact makes dealing with infs and sups very easy.

• Compact subsets are always closed and bounded. The importance of Heine-Borel is that in $\mathbb{R}^n$, the converse is also true.

• Openness and closedness are properties of subsets of a metric space. In every metric space $(X,d)$, the sets $\emptyset$ and $X$ are always both open and closed. (On a related note: sets can be open, closed, both or neither.)

• Cauchy sequences are bounded.

• The actual "formulas" for limsup and liminf are not included. By that I mean: $$\limsup a_n = \inf_{n\geq 1}\sup_{k\geq n} a_k,$$ $$\liminf a_n = \sup_{n\geq 1}\inf_{k\geq n} a_k.$$ This definition is so much easier to work with than "the largest subsequential limit." Perhaps the equivalence of the two concepts can be included as an exercise.

• Let $a,b \in \mathbb{R}$. If $f, g$ are uniformly continuous, then so is $af + bg$, but $fg$ may not be. If $f_n, g_n$ are uniformly convergent, then so is $af_n + bg_n$, but $f_ng_n$ may not be.

• Theorem 7.12 states that uniform convergence preserves continuity. But in fact, uniform convergence also preserves uniform continuity, too, as well as boundedness (which is an un-mentioned step in Exercise 1 of that chapter.) These facts provide additional reason to care about uniform convergence.

• Uniform convergence only makes sense for sequences of functions. Also, it is equivalent to convergence in $C(X,Y)$ with the topology induced by the sup norm. (This is the true content of Theorem 7.9)

• Theorem 7.25 is a weak form of the Ascoli-Arzela Theorem (also un-named). As stated it provides sufficient conditions for the existence of a uniformly convergent subsequence. So what? Well, the real point is that this theorem classifies the compact subsets of $C(X,Y)$ -- much in the way Heine-Borel classifies the compact subsets of $\mathbb{R}^n$. (Admittedly, this bullet point might be a little abstract for most students.) Anyway, this connection is established via Theorem 3.6 which involves sequential compactness (also un-named!), which brings me to:

• Compactness is equivalent to sequential compactness in metric spaces (Rudin only mentions one direction).

Finally, I would suggest that if you are typing up formal solutions that you try to be 100% bullet-proof rigorous. If this course is being used as an introduction-to-proof-writing course, then students would really benefit from seeing what a 100%-correct solution looks like.

(Sorry for all the edits, but I just keep remembering all these extra points!)

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+1 Another excellent and informative answer! –  Amitesh Datta Jul 9 '11 at 3:50
Your first fact is just a consequence of the standard convention that that statement really has a $\forall a\forall b$ implicit around it, so that it stands for $$\forall a\forall b\Bigl(\bigl((\forall\varepsilon>0)(a<b+\varepsilon))\bigr)\implies a\leq b\Bigr)$$ In general, the free variables in a proposition are implicitly universally quantified. –  Mariano Suárez-Alvarez Jul 9 '11 at 4:05
@Mariano: While these are both true, and obvious to both of us now, they were never explicitly pointed out to me. The point isn't so much that it's a deep, essential fact that's missing, but rather a really useful way of framing things, especially for solving problems, and especially for students new to $\delta$-$\epsilon$ proofs. –  Jesse Madnick Jul 9 '11 at 4:10
Because really, half of what makes $\delta$-$\epsilon$ proofs so confusing are the quantifiers! I would guess that many students taking an introductory real analysis course do not consciously know that the negation of [$\exists x$ s.t. $P(x)$ is true] is [$\forall x$: $P(x)$ is false]. –  Jesse Madnick Jul 9 '11 at 4:14
This is a very insightful answer. (I especially like the way you own up to your relative inexperience. By the end of the post, that part made me appreciate it even more.) One tiny point of disagreement: I prefer the "largest subsequential limit" approach to the lim sup, even in my own thinking and especially when explaining to students. Since evidently not everyone feels the same way, the answer seems clear: we should present both approaches and let (and also help, of course) the students decide which one to use in any given situation. –  Pete L. Clark Jul 9 '11 at 4:45

An excellent book for beginners is Stephen Abbott's Understanding Analysis. You might also try A Radical Approach to Real Analysis by David Bressoud

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