Joel David Hamkins 
Proof and the Art of Mathematics [PDF ebook] 

An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.This book offers an introduction to the art and craft of proof writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision. The book is organized around mathematically rich topics (rather than methods of proof), allowing students to learn to write proofs with material that is itself intrinsically interesting. Students will find the early chapters the easiest. Chapter 4 explains the method of mathematical induction, which is used in many arguments throughout the book. Later chapters offer chapter-length developments of major theorems, and the final chapters are more abstract. The book is generously illustrated; an extended chapter on proofs-without-words shows the power of figures and diagrams to communicate mathematical ideas-but also acknowledges the dangers of such an approach. Each chapter includes exercises, and sample answers are provided at the end of the book.