If anyone is interested in how mathematicians think and do proofs there are many helpful books. For example,

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand and others.

How to Prove It: A Structured Approach by Velleman.

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.

Elementary Discrete Math books make for great intro to proofs and math thought:

Discrete Mathematics with Applications by Epp.

Mathematics: A Discrete Introduction by Scheinerman.

There are tons of terrific introductory books on other subjects in math if anyone's interested.