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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1$\begingroup$ Something like "Problem solving strategies" by Arthur Engels (?) $\endgroup$ – Imago May 5 '16 at 17:21
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$\begingroup$ Check this amazon.com/Accompaniment-Higher-Mathematics-Undergraduate-Texts/… $\endgroup$ – MathematicianByMistake May 5 '16 at 17:26
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$\begingroup$ Some possibilities: [amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/…, [amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/…, [amazon.com/Mathematics-Discrete-Introduction-Edward-Scheinerman/…, [amazon.com/dp/0521675995/…, [amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/… $\endgroup$ – Michael Burr May 5 '16 at 17:30
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1$\begingroup$ Since you are preparing for an analysis course I would suggest: [amazon.com/… It is not a technically a book about proofs, it is a book about techniques for handling inequalities, something I struggled with early on my math education: It has many challenging problems, and solutions for you to verify. $\endgroup$ – Luis Vera May 5 '16 at 17:32
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$\begingroup$ Thanks, I just took a look at that Engles pdf and yes! This is a very good looking book. I need to do some problems to get a full feel. If you know of a couple more along this vein but more focused (ie. proofs in Number theory or whatever topic.) . I have enough books on proof with techniques. I need progressive exercises. I am ready to learn about how to think about the proofs from dead simple to advanced undergrad. Eventually I want to teach so I will have the advantage of being empathetic towards students that don't have a full range of techniques. $\endgroup$ – MarkoPolo May 5 '16 at 17:40
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I recommend
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang. (link below)
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
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$\begingroup$ 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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