Sum of Reciprocals of Primes is Divergent/Lemma

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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.

$\blacksquare$