This is it folks, the best there is!
However, it could have been better. I bought the book almost 10 years ago. I am a secondary ed. math teacher and when I left college I was quite upset with myself that I had this fancy math degree and couldn't prove anything. I picked up this book and today I'm working on my PhD in mathematics! This book inspired me to that.
First - What's wrong with the book. Not that there really is anything wrong with the book. I have attempted this book 3 times. I admit, the first two times I stalled (1997 - 2001) when I got to page 119. For some reason I couldn't grip those concepts such as intersecting families, etc. The preface of the book says only high school mathematics is required - that is just flat out wrong. This book is more for undergrads and maybe older fossils like me that have delved into mathematics a bit more than average. Also, like all the other reviews, there is too many exercises with no solutions. What really threw me with that is I didn't know if I was setting the written argument up properly. Sure, on the one hand, it's better to NOT have answers so you strive like a mad person to find them. Yet, it's so frustrating to not know if you did something right. The best approach is to do your best I suppose. After the third try (2004 & 2005) I finally completed the book on my own volition and I'm assuming most of my content is correct.
Velleman describes math so well that I honestly admit, I have a full repetoire of tactics to use to solve mathematical proofs. I don't have the confidence to toy with the big boys yet, like correcting a 49 page proof pertaining to the 'Twin Prime Conjecture' ... but it is SO NICE to UNDERSTAND the arguments! When I took Number Theory, I knew induction well, I know the If P Then Q arguments, it was just a blessing to know what the angle that the provers were using to prove mathematical theorems. I absolutely love this book. The cover is falling off and the pages are wearing out. I'm about to buy a new copy and start all over again. Mastery of this book, will certainly lead to a mastery of proof-writing in mathematics. I totally 100% recommend you buy this book if you are interested in mathematical proofs.