Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.
While it's a good book, I'm not sure it's suited for self study by itself. I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway.
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"Principles of Mathematical Analysis" 3rd edition (1974) by Walter Rudin is often the first choice. This book is lovely and elegant, but if you haven't had a couple of Def-Thm-Proof structured courses before, reading Rudin's book may be difficult.
Thomas's calculus also seems to fit well to your needs, as i myself had used that book and found it more appealing than Rudin's
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The book of Bartle is more systematic; much clear arguments in all theorems; nice examples-always to keep in studying analysis.
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Bryant [1] would be my recommendation if you're fresh out of the calculus/ODE sequence and studying on your own. If your background is a little stronger, then Bressoud [2] might be better. Finally, you should take a look at Abbott [3] regardless, as I think it's the best written introductory real analysis book that has appeared in at least the past couple of decades.
[1] Victor Bryant, "Yet Another Introduction to Analysis", Cambridge University Press, 1990.
[2] David M. Bressoud, "A Radical Approach to Real Analysis", 2nd edition, Mathematical Association of America, 2006.
[3] Stephen Abbott, "Understanding Analysis", Springer-Verlag, 2001.
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You might want to take a look at A Problem Text in Advanced Calculus by John Erdman. It's free, well-written and contains solutions to many of the exercises. These attributes, in my opinion, make it particularly well-suited for self-study. One of the things that I particularly like about the text is the author's use of o-O concepts to define differentiability. It simplifies some proofs dramatically (e.g., the Chain Rule) and is consistent across one-dimensional and n-dimensional spaces.
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I recommend Mathematical Analysis by S. C. Malik, Savita Arora for studying real analysis. A very detailed and student friendly book!
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I really like Fundamental Ideas of Analysis by Reed. It's a friendly and clear introduction to analysis.
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When I was learning introductory real analysis, the text that I found the most helpful was Stephen Abbott's Understanding Analysis. It's written both very cleanly and concisely, giving it the advantage of being extremely readable, all without missing the formalities of analysis that are the focus at this level. While it's not as thorough as Rudin's Principles of Analysis or Bartle's Elements of Real Analysis, it is a great text for a first or second pass at really understanding single, real variable analysis.
If you're looking for a book for self study, you'll probably fly through this one. At that point, attempting a more complete treatment in the Rudin book would definitely be approachable (and in any case, Rudin's is a great reference to have around).
For self-study, I'm a big fan of Strichartz's book "The way of analysis". It's much less austere than most books, though some people think that it is a bit too discursive. I tend to recommend it to young people at our university who find Rudin's "Principle of mathematical analysis" (the gold standard for undergraduate analysis courses) too concise, and they all seem to like it a lot.
EDIT : Looking at your question again, you might need something more elementary. A good choice might be Spivak's book "Calculus", which despite its title really lies on the border between calculus and analysis.