I'm not sure if I understood it correctly, but one of my professors told us that one theorem was proved this way: A mathematician assumed the truth of the Riemann hypothesis and was able to prove a certain mathematical statement. Then a second mathematician assumed the negation of the Riemann hypothesis and was also able to prove the same statement. These two proofs prove that the statement is indeed true. (Does anybody know what this theorem is?)
Another example of such unconventional proof is the proof of the Fermat's last theorem for $n=5$. As I understand it, Sophie Germain showed that if ever there is a solution, one of the integers must be divisible by 5. Dirichlet then proved that if such a solution exists, then the number divisible by 5 must be odd. In the same year, Legendre proved that if such a solution exists, then the number divisible by 5 must be even. Since there are no integers that are simultaneously odd and even, no solution exists.
I also read somewhere that an unconventional way of showing that a set is nonempty is to show that its cardinality is odd (since if the cardinality is odd, it can't be zero).
Do you know of any other very interesting and unconventional proofs that are relatively easy to understand?