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Are there any books in real analysis that explains what goes on in their proofs? I want to self study real analysis. I read through proofs in each of these real analysis books and I'm not understanding anything. (I'm using like 5 books at this point and they still aren't helping.) These books give me the same definitions and meanings but the proofs aren't exactly ticking. I would love if there was a textbook that can show like step by step proofs and why each line of logic works (if there is one). I know I'm supposed to struggle with real analysis but there's no real guide for me. Any pointers and books that meet these requirements? I'm looking for a good book with an intuitive explanation of analysis.

By the way just before we start a debate: It's not a repeated question because i went to this link and looked through all these books also and that wasn't very helpful. Source for those interested: Good book for self study of a First Course in Real Analysis Just for those curious, the five books I'm using are:

  1. Walter Rudin's book (This is like really difficult for me to understand).
  2. Patrick M Fitzpatrick (Advanced Calculus: A course in Mathematical Analysis)
  3. Kenneth Ross' book (Elementary Analysis)
  4. Stephen Abbott's book (Understanding Analysis)
  5. R. Battle's book (Introduction to Real Analysis)
Community
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Kagamine Len
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    Hmm are you looking for explanation of the logical reasoning itself? If so, I recommend you look for an introduction to boolean algebra and natural deduction so that you can understand the inference rules that every math proof must use to derive true statements from the previous ones. If however you're looking for an intuitive explanation of analysis, then I've no recommendations to make. – user21820 Aug 12 '15 at 09:56
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    I'm looking more for the intuitive explanation of analysis. I guess I should add that I'm looking for an intuitive explanation of analysis over the logical reasoning of mathematical statements itself. – Kagamine Len Aug 12 '15 at 09:58
  • Ah yes then you'll have to ask others. The only book I read before was Spivak's calculus, which I found good for me but I'm not sure it meets your criterion of explaining intuition. I second your question because too often the intuition is not given. I haven't seen a textbook explain why we want to consider the Bolzano Weierstrass theorem in the first place. – user21820 Aug 12 '15 at 10:00
  • I have asked the same question, and I have not yet gotten a definitive answer to this myself, but I got a tip about "the real analysis lifesaver" in the thread below. Reading about it, it seems like it might be what you are looking for. https://math.stackexchange.com/questions/2710442/proof-analysis-in-zorns-understanding-real-analysis – KJA Mar 31 '18 at 09:49
  • I would add the book Calculus by Spivak to the list. – littleO Nov 19 '19 at 08:05

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During my Real Analysis course, I used the book "Principles of Real Analysis". This was written by Charalambos Aliprantis and Owen Burkinshaw. It provides detailed proofs and also challenges the reader to think for him/herself. There are also many exercises to practice. I would recommend this book for anyone studying real analysis.

Kagamine Len
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Jan
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I am also a mathematics honours student and I too love real analysis. First you should develop ideas using some standard books, one of which is recommended by Jan, and once you have done this, you can read world class books like Walter Rudin; MATHEMATICAL ANALYSIS of Tom M. Apostol; and also the book ADVANCED CALCULUS of Titu Andreescu. Last three are awesome books for real analysis.