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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.

  • 18
    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.2011-07-14
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    Rudin is **dense** (pun intended); it can take a few readings to digest even just the definitions.2011-07-14
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    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](http://www.amazon.com/Nuts-Bolts-Proofs-Third-Introduction/dp/0120885093/ref=sr_1_3?ie=UTF8&qid=1310621178&sr=8-3) and there are certainly lots more.2011-07-14
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    The problem is that if one is just starting to try to prove things for the first time, it makes it even harder if there's the added intrinsic difficulty of the particular subject, in this case, elementary analysis. So that's why I believe that it is best to first get more comfortable with proofs in the easiest possible subjects and increasingly building up from there.2011-07-14
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    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.2011-07-14
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    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.2011-07-14
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    Analysis is not the easiest material with which to have one’s first encounter with proofs, and Rudin is an exceptionally densely written text. When I last taught such a course, ~30 years ago, I used Robert G. Bartle’s *Elements of Real Analysis*: it covers most of the same ground and is a bit less intimidating, though s still a stiff challenge for a newcomer to proofs. She might find it helpful to dig up a copy of Edward D. Gaughan’s *Introduction to Analysis*: it’s intended to bridge the gap between cookbook calculus and real analysis, and it does a pretty good job with what it covers.2011-07-14
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    She is a rising sophomore at MIT and this book is required for one of the very first proof based classes. The only other intro proof class is one in linear algebra. I suggested that class, but she refused due to the professor's poor ratings. Unfortunately since there are tons of bright math students there (IMO's and what not), there ends up being many freshman who take the analysis class right away. Consequently, there is pressure to dive right into Rudin as soon as possible. Unfortunately, I don't think MIT offers a good track for aspiring mathematicians with no previous math experience.2011-07-14
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    @Scott: See Herbert Wilf's beautiful article [Epsilon sandwiches](http://www.math.upenn.edu/~wilf/website/MAASpeech) — it shows how proficiency with proofs (and proof writing) is the most frustrating part for students in the analysis course. (Mentions Rudin, too.)2011-07-15

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