Proofs with a different flavour.

Enkidu

I've been developing an interest lately in proofs from one area of mathematics that use the techniques and ideas of another area. Of course this is quite common these days, but I particularly like the proof of a classical problem from one area being proved using the basic techniques or even the foundational ideas of another.

An example of what I'm talking about is the proof of the infinitude of the primes using topology.
Topological proof of Euclid's prime theorem.

Does anybody have other examples?

Fremen

Nice wikipedia page:
http://en.wikipedia.org/wiki/Probabilistic_proofs_of_non-probabilistic_theorems

You might find these interesting too, though some of the answers are quite technical:
http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics
http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra