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To start out with, I'm a junior in high school who is intrigued by the rigor of higher mathematics and is currently attempting to self study Volume 1 of Apostol's Calculus. I haven't had any previous experience with proof based mathematics, but I recently stumbled on a book that seems to suit my needs well. The book is Basic Concepts in Mathematics by Zakon, which is supposed to be read before or in conjunction with his two books on analysis. It seems to give a good exposition on set theory and additional topics in a rigorous fashion but without assuming any prior knowledge. I plan to read this alongside Apostol, but was wondering if their were any better books with regard to exposing the reader to proof based mathematics but with a focus on analysis like Zakon's text. Further, the book is free and can be accessed by the following link:

http://www.trillia.com/zakon1.html

I have gone through a small portion of the Book of Proof, but I'm looking for a text less about learning proofs than about applying them and to give me more mathematical maturity. If you have any suggestions for me or thoughts on Zakon's book, it would be greatly appreciated.

user598582
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    One good book to be aware of is Calculus by Spivak. – littleO Sep 29 '18 at 19:30
  • @littleO I've been told that Spivak is less friendly to self-learners than Apostol, hence my studying of Apostol instead. Is this true? I feel intimidated by Spivak because of my inexperience with proofs. – user598582 Sep 29 '18 at 19:37
  • I haven't read Apostol, but I do think that Spivak can be a challenging book. Particularly some of the problems can be quite challenging. I think it's worth looking at Spivak's Calculus to see if the writing connects with you. (By the way, note that I am not recommending Spivak's book Calculus on Manifolds, which is aimed at people who already have a lot of mathematical maturity.) Another thing to consider is that number theory is a great subject for learning about proofs and developing mathematical maturity. You might take a look at an introductory number theory book. – littleO Sep 29 '18 at 19:43
  • @littleO Thanks for the suggestion, I'll definitely take a look at Spivak; in addition to its difficulty, it apparently builds a lot of mathematical maturity, so it could definitely be useful. As too your suggestion on number theory, I haven't had tah – user598582 Sep 29 '18 at 19:49
  • @littleO at recommended to me before but it seems like a good idea. Do you have any suggestions for introductory texts? Thanks again. – user598582 Sep 29 '18 at 19:50
  • I don't really know the best introduction to Number Theory books. This quora thread has some suggestions. Maybe Elementary Number Theory by Rosen is not bad. A Friendly Introduction to Number Theory might be good. – littleO Sep 29 '18 at 20:01
  • @littleO I'll take a look at those. Thank you for your feedback. – user598582 Sep 29 '18 at 20:21
  • Does this answer your question? How do I self-learn undergraduate math? – user1147844 Jun 06 '23 at 07:21

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I was just recommended a few books by @JasperLoy in MSE chat. They're as follows:

  • Calculus

    • Kaplan and Lewis: Calculus and Linear Algebra I and II
    • Spivak [noted in the comments by @littleO]

  • Real Analysis

    • Protter and Morrey: A First Course in Real Analysis
    • Yeh: Real Analysis

I would personaly like to recommend Anton Bivens' Calculus Text to compliment Spivak. Also, Kreyzig Advanced Mathematics is also a useful stepping stone in any math journey. Try out Iridov and also Integral Kokoboken (Cookbook) if you like challenges.

If you're only starting out, I suggest going through various math textbooks like Playing With Numbers and Mathematics X/XI/XII from syllabii set by ICSE, CBSE, GCSE, iB, et cetera.

Best of Luck.

J. W. Tanner
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Nick
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    Sorry for such a late response – user598582 Sep 30 '18 at 07:41
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    I really appreciate the help. Starting off on this kind of journey is difficult because of the lack of experience, so it's always nice for people to help out. Thanks for the answer! – user598582 Sep 30 '18 at 07:43
  • @user598582 I'm sure there is a vast array of much more useful books but the important thing is to just start. It's all down to taking baby steps everyday to attain mastery. – Nick Sep 30 '18 at 12:52