Sum of Reciprocals of Primes is Divergent/Lemma

Theorem

Then:

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

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:

$\ds \map \ln {\ln n} - C$

$>$

$\ds c$

$\ds \leadstoandfrom \ \ $

$\ds \ln n$

$>$

$\ds \map \exp {c + C}$

Definition of Exponential

$\ds \leadstoandfrom \ \ $

$\ds n$

$>$

$\ds \map \exp {\map \exp {c + C} }$

Definition of Exponential

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.

$\blacksquare$