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I haven't read many math books (only the ones required for class and some I picked up on my own) but most all the math books I have read so far leave out important proofs for several Theorems. Sure, proofs should be left as exercises but some Theorems are difficult to prove with lack of mathematical maturity (and I don't like "accepting it" for now and someday learning to prove it).

So is there (or are there) books that have proofs for every Theorem in its specific branch of mathematics? For example, a book on Analysis that has a proof on every theorem (specific to Analysis), from the beginning to the "end"?

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    $\begingroup$ Analysis has an "end"? $\endgroup$ – Lord Shark the Unknown Nov 9 '17 at 6:17
  • $\begingroup$ To be frank, wouldn't this apply to basically any textbook that doesn't cover the foundations of math in detail? So for example, I think the Intro to Set Theory book by Hrbacek and Jech basically starts "from ground zero," but most analysis and algebra books don't talk about set-theoretic justifications.... $\endgroup$ – pseudocydonia Nov 9 '17 at 6:22
  • $\begingroup$ @Lord Shark the Unknown That's why I said "end," because Analysis itself doesn't have one (or does it?) but Analysis books do have an end. Sorry if I wasn't clear on that. $\endgroup$ – Tomás Palamás Nov 9 '17 at 6:36
  • $\begingroup$ @pseudocydonia I'm speaking about Theorems specific to $X$ branch of math such as Intro to Analysis, Intro Abstract Algebra, etc. $\endgroup$ – Tomás Palamás Nov 9 '17 at 6:38
  • $\begingroup$ I would recommend reading "Foundations of Analysis" by Edmund Landau- and then realizing that this is not really what you want. $\endgroup$ – Michael Greinecker Nov 9 '17 at 9:28
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  1. The Real Numbers and Real Analysis by Ethan D. Bloch
  2. Mathematical Analysis I (UNITEXT) 2nd ed. 2015 Edition by Claudio Canuto,‎ Anita Tabacco
  3. Mathematical Analysis II (UNITEXT) 2nd ed. 2015 Edition by Claudio Canuto,‎ Anita Tabacco
  4. Advanced Calculus (Pure and Applied Undergraduate Texts: the Sally Series) 2nd Revised edition Edition by Patrick M. Fitzpatrick (Author)

I can't explain in details why they are good. Just get them and find it yourself.

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  • $\begingroup$ The first one is the best in one variable analysis as far as I know, he covers many important topic that many books often omitted. You can download and see the preface and contents in the springer.com. (warmly warmly recommend!) The 2 and 3, if I remember correct, have solutions for every exercise. 4. makes clear explanation on many topics. I have all these books. I simualtaneously read them whenever I studying analysis. $\endgroup$ – Eric Nov 9 '17 at 6:55
  • $\begingroup$ If you have get one or several of these, and you're satisfied with them, trusting my advice, I can give you 5.~8. ;) $\endgroup$ – Eric Nov 9 '17 at 6:56
  • $\begingroup$ I did check contents of the first book you refer. It is indeed impressive. $\endgroup$ – Paramanand Singh Nov 9 '17 at 10:31
  • $\begingroup$ Hi @ParamanandSingh. Wow! Haven't you heard about this book? I feel that you are the person who knows many books in analysis. I remember you had suggested the Spivak's Calculus and Hardy's book. :) $\endgroup$ – Eric Nov 9 '17 at 15:19
  • $\begingroup$ @ParamanandSingh Bloch deal with the number system very well. He gives both the constructive way from the Peano axioms and the axiomatic way by ordered field + completness, and he explained it crystal clear! And he also clearly explain the 'story' of transcendal functions. By the way, I'm going to get Hardy's book. But it takes some effort to buy it in Asia.. $\endgroup$ – Eric Nov 9 '17 at 15:23

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