# Good examples of proofs in mathematics exemplary of creative reasoning [closed]

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.

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