Our primary textbook is An Infinite Descent into Pure Mathematics by Clive Newstead. This book is is freely available (choose version 0.4 the “stable download”). We shall follow this book closely. Note that this book is under development and the author would like feedback, constructive suggestions or comments based on your experience reading the text.
An optional secondary resource is How To Prove It: A Structured Approach by Daniel J. Velleman. This book was used as the course text in the previous iteration of this class. We will not follows this book, but you can find in there a wealth of examples and exercises if you need more practice.
If you are interested in exploring proofs in more diverse mathematically rich topics beyond metric spaces, I recommend the following books. They have plenty of exercises to keep you busy.
J. D. Hamkins, Proof and the Art of Mathematics, MIT Press, 2020