Sum of Reciprocals of Primes is Divergent

Theorem
$$\sum_{p \in \mathcal{P}}^n \tfrac{1}{p} > \ln ( \ln ( n)) - \ln(\tfrac{\pi^2}{2})$$,

$$\lim_{n \to \infty} \left({ \ln ( \ln ( n)) - \ln(\tfrac{\pi^2}{2})}\right) = + \infty $$

Proof
Zelmerszoetrop has this written up from a final exam last semester, I'll post it up today