Very amusing. Here's a proof by contradiction, for the record. Assume that there are only a finite number, n, of prime numbers. There are at most n^3 distinct odd numbers that can be formed as a sum of three of those primes. But n^3 is finite, and so the number of odd numbers must be finite, which is, of course, false, and so our original assumption must have been false. QED

There are some hilarious variations on this theme (using difficult theorems to prove simple facts) here: http://mathoverflow.net/questions/42512/awfully-sophisticate...