Twin Prime Proof Proffered
You can find a proof of this here for example. The idea is to use Euler's product formula for the zeta-function and the well-known fact that the harmonic series diverges.
Interestingly enough, it is known that the sum of the reciprocals of twin primes converges (and the sum is approximately 1.90216054). Of course, if there is only finitely many twin primes, this statement is trivial.