Reading list

There are very many books devoted to introducing mathematical proofs. The best book to learn the material is the one you find the best. Here is a list of other introductions to proofs, in no particular order:

R. Maddox, Mathematical Thinking and Writing: A Transition to Abstract Mathematics.
D. Smith, M. Eggen and R. St. Andre, A Transition to Advanced Mathematics.
D. Solow, How to Read and Do Proofs.
D. Velleman, How to prove it, a structured approach.

Here are some more avanced books which you might find interesting and use the material in this class:

R. Haggerty, Fundamentals of Mathematical Analysis.
D. Burton, Elementary Number Theory.
K. Devlin, Sets, functions and logic, an introduction to advanced mathematics.

Finally, you might consider reading the following book for fun, which in part contains an informal introduction to the ideas behind Gödel's incompleteness theorem, presented through a sequence of conversations between Achilles and a Tortoise :

D. Hofstadter, Gödel, Escher, Bach: an Eternal Golden Braid