Prove that 'pi' is irrational.

This is not a one-liner proof, but also not so bad. The trickier part is to prove that pi is transcendental. (i.e. not a root of any polynomial with integer coefficients)

There's a relatively simple proof of the irrationality of pi up at:

http://pi314.at/math/irrational.html

It is a contradiction proof due to Niven. Not exactly the sort of thing one would come up with off the top of one's head, but not so hard to understand.