Experiences with Rudin? 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. 
 A: 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.
A: 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,
A: 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.
A: Your friend does not need to worry. Even (real) mathematicians get frustrated reading others' work sometimes, and there are many I know who find Rudin to be difficult to read, especially if it is the first exposure of mathematical textbooks. 
This being said, there are several things that need to be addressed that might help your friend.
The first thing is, why is she reading Rudin? Rudin is written with a special focus on proofs. However, in many fields, there are other impeding matters other than proofs. For example, most of my friends do not need to know how Riemman integrals are defined, but they do need to understand how to calculate derivatives, integral etc. If in any case your friend does not need to understand such proofs, then probably Rudin is not the best choice at the moment. 
A second thing is that the writing style of Rudin quite peculiar (although not unique) to mathematics. It largely comprises of definition, lemma, theorems, proofs and examples, while lacking sometimes, for example, motivations and intuitions. One consequence of this style is that the writing is rather dense, and certain concepts only start to manifest their importance long after they are introduced. Another consequence is that proofs are rarely given intuitions before they are presented. All and all, this means the readers cannot simply read the books line by line as with many other textbooks, but have to seek attentively the intuition behind certain definitions and propositions. This also requires the readers to look back, and try to organize the material in their own way so that what is written on the book is no longer just disconnected facts. Unfortunately, doing so requires time and training and the process is rather personal. Luckily there are many resources to help, such as those pointed out by the answers before. I have also found that discussing with friends (with similar mathematical maturity) extremely useful.
Finally, there are sometimes certain writing styles that are particularly easier for certain people to understand while not others. Personally, I have found this much easier read to me, despite that fact that it is in certain sense more abstract. Again, this choice is rather personal so it would be worthwhile for her to first select the books that she is most comfortable with.
Hope these help
A: I think the first is to know the basic of the logic.  e.g. $A \Rightarrow B$ equivalent to $! B \Rightarrow !A$, the exchange of order on $\forall$ and $\exists$. This is important. They can help you to convert the intuition under the statements and the definitions into the rigorous proof.
A: Anyone that can just casually read Rudin's Principles and understand it all without effort is an outlier IMO. For me, sometimes I spend an hour on a single sentence. So the frustration was there for me too but it went away when I set different goals for myself. Initially I wanted to read a certain number of pages in a certain time period - now, I no longer care how long it takes, I see each little proof as an individual accomplishment and a mathematical marvel that deserves to be studied in detail. Completely digesting the material takes a long time but is very rewarding in the end. 
A: 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. 
A: It took me a long time to learn Chapter 2. I was learning to think mathematically and digest proofs. But I can only say that the effort proved very worthwhile. This chapter is in some ways the hardest chapter in the book, but the most useful. Give it everything you've got! You will not regret a single minute struggling with it. I worked all the problems in that chapter, and can even now recall many surprising moments, understanding the topology of the Cantor set, and proving the Baire Category Theorem. In hindsight, an amazing chapter, and an amazingly well conceived book. It will make a mathematician out of you if you give it the careful attention that it deserves.
To the teacher, urge patience and perseverance, and use the Socratic method to teach. Solutions to all the exercises are available online these days. But when I did these problems around 30 years ago, I assumed I was on my own, and treated working through Rudin as the most important thing in my life.
