Experiences with Rudin?

Question

So I am trying to tutor a friend in analysis. This is her first time with proofs. We are on chapter 2 – the topology chapter – of Rudin's Principles of Mathematical Analysis and she is extremely frustrated, mainly because she expects herself to learn at a more rapid pace than is occurring (although she is doing fine imo). When I was learning the material, I recall Rudin taking a long time, as I presume it was for many first timers.

So what are you guys' experience with Rudin? How long did you spend on chapter 2? Is there anything that you found useful to help you get through the book?

I am hoping that if she sees that the math community finds the material/book challenging (assuming you do), she will feel more comfortable.

Answer 1

I'd say except if your friend has a real potential which is almost visible from space, Rudin is probably not the best starter. There are many friendly introductions to analysis which are more intuitively appealing for starters, i.e. for instance, when defining the notion of a limit, instead of just shooting an abstract definition and proving theorems, they could make a drawn explanation of the epsilon-delta definition of a limit, or for the pre-image definition of continuity, making a drawing of what happens when you take the pre-image of an open interval of a continuous function. Very often, it is important to have intuition in analysis in order to prove things correctly, because without that, you are blind. Blind men have done much, but I believe they had wished to see. ^^

I have personnally read the Rudin up to chapter 3~4 after having done one course in basic analysis. Having done a few exercises in it has risen my level of understanding in proofs greatly, but had I done it before that analysis course, I would've been killed. =)

Hope that helps,

Answer 2

Rudin requires patience! The writing is very clear, but it's the kind of stuff that takes a lot of work to absorb.

In both 9th grade and 11th grade I took courses that spent probably half the year on proofs: what they are and how to discover them and write them. That was essential to being able to cope with Rudin. I don't know that I could have done it otherwise.

Answer 3

I agree with Adrián that Rudin, and analysis generally, is not a good first exposure to proofs. There are at least three subjects I can think of off the top of my head that are much more accessible for such a thing:

• Elementary number theory
• Elementary graph theory
• Elementary combinatorics

In these subjects the objects one is proving facts about are much easier to grasp intuitively. I don't know good references at the introductory level off the top of my head, but you might try telling your friend to browse the Art of Problem Solving books.

Answer 4

Rudin was also my first exposure to proofs, and of all the chapters, Chapter 2 took the longest by far. (Other long chapters were Chapters 3 and 7.) I think this was because in transitioning from Chapter 1 to Chapter 2, there is a sudden spike in abstraction. But once Chapter 2 is over and dealt with, the amount of abstraction levels off and, I think, becomes more manageable.

As I see it, Rudin's terseness provides two annoying obstacles to the novice reader, especially in Chapter 2: (1) the lack of examples, and (2) the lack of facts. By "lack of facts" I mean, for instance, how Rudin shows that compactness implies limit point compactness, but doesn't mention that the converse is true.

See also my answer to this related question.