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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.

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    $\begingroup$ Everybody starts with a different level of background and learns at their own pace in their own way. Tell her not to worry about it. It's not a race. $\endgroup$ – Qiaochu Yuan Jul 14 '11 at 4:58
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    $\begingroup$ Rudin is dense (pun intended); it can take a few readings to digest even just the definitions. $\endgroup$ – The Chaz 2.0 Jul 14 '11 at 5:21
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    $\begingroup$ Scott, if this is her first time with proofs then maybe it would be easier for her to start proving not things from analysis, but from elementary set theory and related topics, like easy number theoretic facts. There are lots of books that introduce the students to some basics of elementary logic and teach some of the basic proof techniques or strategies. For example there's this book and there are certainly lots more. $\endgroup$ – Adrián Barquero Jul 14 '11 at 5:30
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    $\begingroup$ I second Qiaochu's comment. There are people who might read Rudin in a year, people who might read Rudin in a month and even people who might be capable of reading Rudin in a week. However, the point is that Rudin is and can be used as one's first exposure to rigorous and theoretical mathematics. Therefore, it is natural that the material might seem abstract at first. Nonetheless, one will become faster at learning mathematics as one progresses through the subject. $\endgroup$ – Amitesh Datta Jul 14 '11 at 5:45
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    $\begingroup$ Of course, Rudin is elementary in the sense that one does not need much background to begin reading the text. In fact, a familiarity with only the rational numbers suffices in theory. In practice, one would want a student to have had at least a high school calculus course. $\endgroup$ – Amitesh Datta Jul 14 '11 at 5:47
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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.

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    $\begingroup$ Maybe... I think a lot depends on the student's interests and motivation. If you don't find a certain subject well-motivated, or simply lack interest in the subject, then the proofs will be a pain. On the other hand, if one has a high enough interest, then although it might be challenging, reading and writing proofs will ultimately prove rewarding. $\endgroup$ – Jesse Madnick Jul 14 '11 at 19:36
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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.

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  • $\begingroup$ He does explicitly mention limit point compactness implies compactness, but that he leaves it as an exercise for the reader to show this (Chapter 2, #26 I believe is the actual exercise). $\endgroup$ – user1770201 Apr 2 '13 at 2:07
  • $\begingroup$ I meant that Rudin doesn't explicitly mention it in the body of the text. I mean, he could have at least made reference to Exercise 26. $\endgroup$ – Jesse Madnick Apr 2 '13 at 2:24
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    $\begingroup$ He makes reference in the text though. $\endgroup$ – user370967 Jul 8 '18 at 20:44
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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.

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    $\begingroup$ Your high school must have a very nice curriculum...unless you took these courses at a college. $\endgroup$ – user13255 Jul 14 '11 at 16:23
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    $\begingroup$ ....lest anything be misunderstood: I never heard of Rudin in 9th or 11th grade. But when I later encountered Rudin it was because of my experiences in 9th and 11th grades that I was able to write proofs called for in the exercises and understand proofs in the text. $\endgroup$ – Michael Hardy Jul 19 '11 at 7:32
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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,

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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.

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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

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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.

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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.

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