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Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be recent) that are good because they employ clever reasoning, and which might serve as good examples of what kind of reasoning (or approaches) is prominent in higher mathematics.

Thanks to all in advance!

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closed as not constructive by Thomas Andrews, Serkan, Sasha, Shuhao Cao, tomasz May 16 '13 at 0:35

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You should really be mor specific - this is a remarkably vague question. Samples of what you are looking for would help. –  Thomas Andrews May 15 '13 at 22:35
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Overly broad, or at least CW –  mrf May 15 '13 at 22:35
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You may enjoy 'Proofs from THE BOOK' with preview if you don't already know it... –  Raymond Manzoni May 15 '13 at 22:43
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Proof that $\sqrt{2}$ is irrational, Proof that there are an infinity of primes $\equiv 1 \pmod8$, Proof of Pivot Theorem, Proof of the law of quadratic reciprocity, Proof of An Euler walk on an finite connected graph exists if and only if the degree of every vertex is even, Proof of Cauchy Functional Equation and Proof of Intermediate Value theorem are some of my personal favorite relatively simple proofs. –  John Marty May 15 '13 at 23:11
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@ThisIsNotAnId: With many suggestions by Erdös himself this book is hard to replace ! –  Raymond Manzoni May 15 '13 at 23:12
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