I tried to prove the divergence of the prime reciprocals as a challenge and I think I came up with quite an intuitive argument using Borell Cantelli, but maybe not rigorous.
For two primes $p_n>p_m$ and a large integer $N >>p_n>p_m$, the probability of the event $P(E_n)$ that that $p_n$ divides N is $1/p_n$, and independent of $E_m$ the event that $p_m$ divides N.
If we assume by means of contradiction that the prime reciprocals do converge, than $$\sum_{n=1}^\infty P(E_n) < \infty $$ And by Borel Cantelona, we have that only finitely many $E_n$ happen simultaniously with probability 1. It can then be seen quite easily (or does this need further clarification)? that the probability of none of the $E_n$ must be larger than 0. Thus there is some $\epsilon > 0$ so that for large N, the probability that no primes divide $N$ is at least $\epsilon$. That means that N is a prime with probability $\epsilon$ and thus the primes have positive natural density of $\epsilon$ , but then their reciprocals should diverge; a contradiction.
Is this a valid proof and reasoning ? Thanks for reading it!