Sum of Reciprocals of Primes is Divergent

Theorem

 * $\displaystyle \sum_{\substack {p \in \Bbb P \\ p \le n} } \frac 1 p > \ln \left({\ln \left({n}\right)}\right) - \ln \left({\frac {\pi^2} 2}\right)$


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

Proof of Limit
Observe the following simplification:

Fix $c \in \R$. It suffices to show there exists $N \in \N$, such that:


 * $(1):\quad \displaystyle n \ge N \implies \ln \left({\frac {2 \ln n} {\pi^2} }\right) > c$

Proceed as follows:

Now, obviously, any $N$ with $N > \exp \left({\dfrac {\pi^2 \exp c} 2}\right)$ satisfies condition $(1)$ by Logarithm is Strictly Increasing and Concave.