# Sum of Reciprocals of Primes is Divergent/Lemma

## Theorem

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

Then:

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

## 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:

 $\displaystyle \ln \left({\ln n}\right) - C$ $>$ $\displaystyle c$ $\displaystyle \iff \ \$ $\displaystyle \ln n$ $>$ $\displaystyle \exp \left({c + C}\right)$ Definition of Exponential $\displaystyle \iff \ \$ $\displaystyle n$ $>$ $\displaystyle \exp \left({\exp \left({c + C}\right)}\right)$ Definition of Exponential

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

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

Hence the result.

$\blacksquare$