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After taking an introductions to proofs course and abstract algebra, I have been trying to study from Rudin's Principles of Mathematical Analysis. Unfortunately, I still find it very very difficult to read through Rudin let alone do the exercises. Sometimes I spend an entire week trying to understand a proof but get nowhere.

At this point, would it be advisible to read a more introductory text in analysis or continue to work through Rudin? Or try reading some more abstract algebra?

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I agree with Chris's recommendation below. I would add: after trying some other analysis books, revisit Rudin, only just Chapters 3-6. One of the reasons Rudin is so difficult to get into is that the first chapter is extremely heavy on axiomatics, and the second chapter contains much of the general theory of metric spaces. Very little of this is actually needed for Chapters 3-6, which are approximately the content of the the usual "first course in analysis". Provided you have some introduction, these chapters are quite readable. You can always revisit the other stuff later. –  leslie townes Apr 24 '12 at 1:20

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Rudin can be a bit terse. Here is an alternative: Introduction to Real Analysis by Bartle . It's a good book. It dosen't cover Dedekind cuts though, if that's what you're struggling with (you said that you just started reading it, and the first part of the book is on Dedekind cuts). Anyways, read what you like. Some people like algebra more than analysis. Some are the other way around. Don't think that you have to read from specific famous books. People have learning styles, and this is one reason why we have many different books on the same subject.

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I have some fairly extensive experience with Rudin, and I had some similar issues to you.

When I was in high school I visited a book store and purchased Rudin saying "I'm good at calculus, this must be the next step!" Of course, I was sadly mistaken. I managed to muddle through a fair amount of the book doing about half the exercises (how much I actually understood is up for debate). It was really rough. Looking back though I am able now to identify the things that made the book very difficult for me.

First and foremost though, and this goes with almost ALL beginning upper level mathematics, was that I was not really well acquainted with the difference between a "function" and a "map". In other words, I was not used to thinking about a function as being defined on anything other than it's maximal domain and it's codomain being just it's image. So, notions of countability and the lot got me all kinds of confused. In the same vein, I was unable to follow a lot of proofs (the one that sticks in my head is that the image of compact under continuous is compact) because the idea of manipulating preimages via unions and the sort was foreign to me. If you want to be successful at Rudin I suggest you get very comfortable with the basics of set theory, as laid out in the beginning of a book like Munkres.

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When you got stuck on reading a proof in Rudin did you just simply skip it and read ahead? Or did you just spend all your time trying to understanding it? –  Student Apr 25 '12 at 3:19

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