Experiences with Rudin?

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.

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.