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.
Best Answer
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.