Define to be the primes . For a given (sufficiently large) let be the primes .
Notice that .
Therefore, we see that .
By the fundamental theorem of arithmetic any number be be written as where .
It follows that,
Since the harmonic series diverges it means .
We have shown,
Remember that .
What this means is that if then would be bounded.
This is impossible, it is not bounded.
Thus, the sum of prime reciprocals must diverge.