Sum of Reciprocals of Primes is Divergent/Proof 1

Proof
By Sum of Reciprocals of Primes is Divergent: Lemma:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\map \ln n} - \frac 1 2} = +\infty$

It remains to be proved that:
 * $\displaystyle \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - \frac 1 2$

Assume all sums and product over $p$ are over the set of prime numbers.

Let $n \ge 1$.

Then: