# Advanced undergraduate(?) Real Analysis book which is concise and lots of interesting problems

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 e.g.baby 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 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 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. –  ramanujan_dirac Jul 30 at 16:04
You may find interesting "Real Analysis and Probability" Ash and "Real analysis" by Folland. –  Ilya Jul 30 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 at 10:04
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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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