Sum of Reciprocals of Primes is Divergent

Theorem
Let $n \in \N: n \ge 1$.

There exists a (strictly) positive real number $C \in \R_{>0}$ such that:


 * $(1): \quad \displaystyle \sum_{\substack {p \mathop \in \Bbb P \\ p \mathop \le n} } \frac 1 p > \map \ln {\ln n} - C$

where $\Bbb P$ is the set of all prime numbers.


 * $(2): \quad \displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = +\infty$