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I have gone through the other book recommendations on Real Analysis, but I think my requirements and background is slightly different. I am a Physics undergrad teaching myself Pure math. My journey is pure math has been highly non-linear. I have studied Groups and Rings (Dummit and Foote), some Commutative algebra (Atiyah and MacD ~3 chapters), and some representation theory(Fulton and Harris). I am looking for a challenging enough book for Real Analysis. It should cover the material in for Rudin, but I am thinking of something more concise but deeper, which has maybe more interesting and difficult problems.

I have done a course on Real Analysis taught from Bartle and Sherbert (I hope this text is not very unknown), but I wish to revisit the material and learn, maybe upto what a standard math undergrad is supposed to know, and also to develop my problem solving skills.

Please feel free to close down the question.

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Personally, whenever I'm asked for any kind of Real Analysis study book I cannot more highly recommend "Introduction To Real Analysis" by William F. Trench. Google the title and author and a pdf copy exists online (not that I necessarily endorse this behavior). – Autolatry Jul 30 '13 at 15:50
Why not read Rudin's book? It's challenging, concise, and has nice problems. I don't understand why you don't want to read it, because it fits all of your criteria. – Potato Jul 30 '13 at 15:56
@Potato: Ohh. I didn't have a good look at Rudin, but I assumed it overlaps too much with the last Real Analysis text I read, Bartle and Sherbert. I will have a look. – user23238 Jul 30 '13 at 16:04
You may find interesting "Real Analysis and Probability" Ash and "Real analysis" by Folland. – Ilya Jul 30 '13 at 16:08
I also recommend baby Rudin. I come from an analysis background (real analysis, complex analysis, functional analysis, topology). I find Rudin's exercises useful, interesting, and challenging. The first chapter may be off-putting for some - it is probably the most difficult chapter if you don't have a good background in pure maths - but it isn't necessary to understand all of the first chapter, e.g. the construction of the reals from the rationals. Personally I like Rudin's style, which I find to be thorough and meticulous. Imo he conveys the main points and theorems with excellent clarity. – Adam Rubinson Jul 31 '13 at 10:04

Have a look at Charles Chopmon Pugh's book on real analysis. This is one of the best books that I know of. It has an intuitive approach which is necessary for a physicist, yet, it doesn't sacrifice rigor for making arguments simple. It has some very good problems. I particularly like the chapter on topology. One of the advantages of this book over baby Rudin is that it discusses both open cover compactness and sequential compactness. I think the best part about this book is that you can learn a lot from this book with the least prerequisites I think.

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When I was doing Real Analysis, the text was Stephen Abbott's Understanding Analysis, but my lecturer wrote his own set of notes which he has subsequently made into a book: Lectures on Real Analysis by Finnur Lárusson. I highly recommend the book. It starts with the basic axioms of $\mathbb{R}$ and finishes with metric spaces. It has over $200$ problems, some of which I have done and found quite challenging and enlightening.

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Since you are a physicist, I would point out that unlike mathematicians, physicists are allowed to use infinitesimals. Therefore I would heartily recommend this book:

Vakil, Nader Real analysis through modern infinitesimals. Encyclopedia of Mathematics and its Applications, 140. Cambridge University Press, Cambridge, 2011.

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I like Kolmogorov and Fomin "Introductory Real Analysis" - which gives lots of examples and has plenty of good problems. But I'm not sure what kind of problem you are looking for.

If you are after challenging integrals and limits etc Hardy's "Pure Mathematics" has lots of those. I also think Apostol's Mathematical Analysis has a mix of challenging problems.

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