Proofs

Below is a list of reviews of books on introductory proofs in mathematics. Most require few prerequisites, if any.

Mathematical Proofs: A Transition to Advanced Mathematics
Chartrand, G., Polemni, A. D., and Zhang, P. (4th Ed. 2017). ISBN 9780134746753.
No prerequisites.

Very clearly written, neither too slow nor too fast. Modern notation and low levels of hand waving. Contains worked examples and outlines of proofs, full solutions to many exercises. Later chapters serve as introduction to various topics.

The Art and Craft of Problem Solving
Zeitz, D. (2nd Ed. 2006). ISBN 9780471789017.
Assumes some basic calculus and comparable fundamentals. Anyone with an open mind should be able to profit from it.

The book is really well written, and focuses a lot on the process of solving a problem in addition to actual problem solving. It has a lot of problems after theory. It focuses on the basics of problem solving, and has chapters on problem solving strategies, cross-over tactics, combinatorics, number theory, geometry and some analysis and linear algebra.

How to Prove It: A Structured Approach
Velleman, D. J. (3rd Ed. 2019). ISBN 9781108439534.
No prerequisites.

A great introduction to proof oriented mathematics (higher/pure mathematics), for people who have never done that kind of thing before.

Book of Proof
Hammack, R. (3rd Ed. 2018). ISBN 9780989472128.
No prerequisites, though a few examples come from calculus.

A light but decent overview of proof writing and how sets and logic work at a very basic level. It is great for anyone who wants a swift introduction to mathematics. Could easily be used in conjunction with introductory texts in linear algebra or analysis as a sort of companion to enhance proof writing from the start.

Introduction to Advanced Mathematics Course notes
Aluffi, P. Free Download.
No prerequisites.

Course notes for a class taught at Florida State University, so each chapter corresponds to one class taught. Very nice writing style and superb presentation with many pictures and good coverage, where the first half corresponds to ‘must know’ things for everyone looking into doing advanced mathematics and the second half covers some rather advanced topics more in-depth (Dedekind cuts, cardinalities, topology). Includes good homework problems.