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What are the best introduction books to mathematical proofs in undergraduate mathematics? I know of "Proofs from the Book" by M. Aigner and G. Ziegler, but also need one that shoots to analysis kind of proofs.

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The "proofs from the book" are wonderful, but not ideal as a first introduction, I think. –  Goldstern Oct 22 '13 at 22:23
Analysis is not good as introduction to the proofs. Euclidean geometry is way better. –  Anton Petrunin Oct 22 '13 at 22:37
You might try Kevin Houston's 'How To Think Like a Mathematician', a gentle introduction to proof and to reading and understanding mathematics. –  Marius Kempe Oct 23 '13 at 0:09

5 Answers 5

How to prove it, by Daniel Velleman.

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I second Martin Goldstern's recommendation of Daniel J. Velleman's How to Prove It: A Structured Approach (New York: Cambridge University Press, 1994). –  grshutt Oct 23 '13 at 3:00

Analysis with an Introduction to Proof, by Steven R. Lay

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I am a bit confused about what do you need. Proofs from the book is (for me) about mathematical beauty. It is a very nice book, but I would not recommend it as an introductory book as first you have to see a lot of not-that-nice mathematics to really appreciate the beauty presented there.

If we assume that by "introduction to proofs" you mean that you know about basic stuff, and need a book where useful 'analysis kind of proofs' are presented i would recommend:

'Problems and Theorems in Analysis I&II' by George Pólya and Gabor Szegö.

If my assumption is false, and you really want to start learning analysis I would not recommend these books.

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I am taking introduction to Analysis,the book we are using,Principles of Mathematical Analysis(Rudin) is not so doing the job.Then there is this other Mathematics course which the professor picks any topic of interest to show the beauty of mathematics(where he refers to "Proofs from the Book").So,what I need is a book which clearly explains proofs and not just the leave-the-rest-for-reader-kind-of-book –  Emma Oct 22 '13 at 23:02
Then i would not recommend these books. They are similar to the 'Proofs from the book' in the sense that they intend to present beauty. But you need to know a lot of basic stuff to benefit from reading these. (Although it is worth it after a year of calculus, if you are still interested.) –  Daniel Soltész Oct 22 '13 at 23:08

The William Lowell Putnam Competition is an excellent example of undergraduate proofs which are fun to solve. You can buy their book or find their problems online. I'm 14 years old but I like it since it gives me a nice look at a variety of elegant proofs which helps high school proof-writing as well. I highly recommend it!

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Not a book, but a nice and (very) brief intro is provided by Joel Feinstein's camcasts (these and these) and the accompanying materials. On the same site, he has also posted an introductory course on real analysis (course materials and recordings of the lectures included) where you can find further "analysis kind of proofs".

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