Velleman's How to Prove It greatly helped my ability to construct set theoretic proofs, which better prepared me for Spivak's calculus and Baby Rudin. Hamkins' Proof and the Art of Mathematics is designed as a a good, less set-theory heavy, introduction to proof writing that leads more naturally to analysis. OpenStax books are FREE.
Surprised to see Velleman's book so far down. It taught me that proofs are fun and do not in general require clever tricks. As a bonus, it provided plenty of practice with foundational objects such as sets, relations and functions. All this made me much better at doing mathematics and prepared to texts in real analysis, CS, algebra.
I have been going through Velleman, and it has been significantly helping me understand various CS papers and books, for example, I struggled understanding through proof outlines in PFPL, but working through just part of this book has helped.
I have had life things interfere with my learning now for the past month or so, but I hope to get back to it soon.
