Sum of Reciprocals of Primes is Divergent/Lemma

Theorem
Let $C \in \R_{>0}$ be a (strictly) positive real number.

Then:


 * $\displaystyle \lim_{n \to \infty} \left({\ln \left({\ln n}\right) - C}\right) = + \infty$

Proof
Fix $c \in \R$.

It is sufficient to show there exists $N \in \N$, such that:


 * $(1): \quad n \ge N \implies \ln \left({\ln n}\right) - C > c$

Proceed as follows:

Let $N \in \N$ such that $N > \exp \left({\exp \left({c + C}\right)}\right)$.

By Logarithm is Strictly Increasing and Strictly Concave it follows that $N$ satisfies condition $(1)$.

Hence the result.