Sum of Reciprocals of Primes is Divergent/Lemma

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

Then:


 * $\displaystyle \lim_{n \mathop \to \infty} \paren {\map \ln {\ln n} - C} = + \infty$

Proof
Fix $c \in \R$.

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


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

Proceed as follows:

Let $N \in \N$ such that $N > \map \exp {\map \exp {c + C} }$.

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

Hence the result.