"Comprehensive Proof" Math books? 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"?
 A: *

*The Real Numbers and Real Analysis
by Ethan D. Bloch 

*Mathematical Analysis I (UNITEXT) 2nd ed. 2015 Edition by Claudio Canuto,‎ Anita Tabacco

*Mathematical Analysis II (UNITEXT) 2nd ed. 2015 Edition
by Claudio Canuto,‎ Anita Tabacco

*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.
A: As for analysis, Vladimir Zorich's Mathematical Analysis (I and II) and Terence Tao's Analysis (I and II) are very hard-core in the sense that they are totally proof-based  and very comprehensive, and you can always find useful theorems or lemmas proved in these two textbooks. Zorich's Analysis is also famous for both its broad coverage of the application of mathematical principles to physics, and the difficulty of the review problems. It can benifit you greatly if you put great efforts into thinking and solving them independently.
As for linear algebra, I highly recommend the textbook "Linear Algebra 5th Edition" written by Stephen Friedberg, Arnold Insel, and Lawrence Spence. It is true that this books is very expensive (230 Canadian Dollars), but it can explain every concept very clearly and thoroughly prove the theorems. This is also the textbook of my school for elementary algebra. The review problems of this book are also very challenging.
As for other branches of mathematics, Thomas Jech and Karel Hrbacek wrote a book called "Introduction to Set Theory", which uses axiomatic ways to construct set theory and of course is based on proofs of the theorems; also, Jech himself wrote a brick-like book called "Set Theory: The Third Millennium Edition, revised and expanded", and this book covers almost all the important topics  in set theory, from its axiomatic foundation to its most current research outcomes, and is of course proof-based.
This is all that I know. I hope it may help you.
