Book of Proof

I just started reading Book of Proof by Richard Hammack and I agree it's an amazing book

Mathematik für Ingenieure und Wissenschaftler I, II and III from Lothar Papula (in German). The solutions are detailed, making it perfect for self-studying. Book of Proof by Richard Hammack. A great introduction to proofs in mathematics. The book is available free online [0], but also I bought the physical version because I really enjoyed it. [0]https://jdhsmith.math.iastate.edu/class/BookOfProof.pdf

Supposing you have at least high school mathematics under the belt: 1. Book of Proof by Hammack [1] 2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra) 3. Linear Algebra by Insel, Spence & Friedberg 4. Understanding Analysis by Stephen Abbott 5. Tom Apostol Calculus Vol 2 These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics. Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university. [1] https://www.people.vcu.edu/~rhammack/BookOfProof/ [2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTkC6qJF1sVh3_Hx7aL6FPd0IN) (the first video is a bit blurry, it gets better)

> You can work through the concepts and delude yourself into thinking you understand something when really you're just hand waving it I keep seeing this and I don't quite understand how this one works. If I were studying math on my own (which I've done and still do), I'd do the following: 1. Pick a book. Say, Rudin's Principles of Mathematical Analysis[0]. Read a section, then attempt problems. Pick a problem. Say, "prove {1/n: n is natural} U{0} is compact directly from the definition(not using Heine-Borel)". It's guaranteed your "proof" is not a proof. 2. Compare your solution to existing solution manuals or ask a question on MSE[1]. Since the given book by Rudin is super-massively famous, each question has probably been asked/answered about a bagillion times each on MSE, so just searching MSE alone would likely to spit out many answers to your questions. People on MSE will tell you exactly why your "solution" is wrong and where you tripped up. Sometimes even the clarifying answers are hard to understand. But then you can ask new questions, think more, correct your misconceptions until it all finally clicks. Do that with all the rest of the problems[2].…