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I am looking for a book that teaches proofs and the book has many exercises from very simple to more difficult? I have noticed with most math books, they seem to leave out pieces too soon before the reader gets a chance to work through lots of examples of a specific level before progressing to the next level. I want more structured approach so that after a couple dozen proofs of class "A", I want to advance to class "B". I cannot overemphasize my need for exercises. BTW, this is in preparation for an analysis course.

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I recommend

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang. (link below)

http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094/ref=sr_1_1?ie=UTF8&qid=1462468981&sr=8-1&keywords=mathematical+proofs+a+transition+to+advanced+mathematics

The nice thing about this book is that it teaches proofs from the ground up. You learn basic proof techniques in a variety of mathematical areas including analysis, abstract algebra, and number theory. The book is very simple to read and has lots of great exercises, including very easy ones as well as more challenging ones. It presupposes no background in any of the aforementioned fields.

The nice thing is that you can get a taste of the different kinds of mathematical proofs (including the $\epsilon-\delta$ proofs you'll be seeing a lot of in analysis) in a simple, clear, informative setting. I.e., the things that you learn to prove and are proved for you are not that advanced, but the help illuminate what really goes on in a proof.

The link I put is to the 3rd edition, but I see that the 2nd edition is available on amazon at a much more reasonable price.

Good luck!

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  • $\begingroup$ I've been eye balling this book for weeks. The content is good. One of the more interesting points is that one of the complaints of a reviewer is in reference to having too many intermediate details. $\endgroup$ – MarkoPolo May 5 '16 at 18:01
  • $\begingroup$ Most worked out proofs in the book definitely contain a lot of intermediate details. The point of a book like this is to ensure that the reader understands the logical structure behind various types of proof. As a result, a lot of the details are worked out to ensure that the reader gets stuck on non-proof-related issues as rarely as possible. Like I said in my answer, the content of the book (ie, what is proved in the book) is not very advanced, the focus is on the structure of the proofs. If one understands the intermediate details, they can be skipped, but I think this is a good starting pt $\endgroup$ – ervx May 5 '16 at 18:25

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