(open) Divergent series of inverse primes

[fresh_42]

Show by contradiction that
$$
\sum_{p\in \mathbb{P}}\dfrac{1}{p} =\sum_{p\;\text{prime}}\dfrac{1}{p}
$$
diverges. Which famous result is an immediate corollary?

Spoiler

Skeleton (proof by Pál Erdös):

1. $$\underbrace{p_1<\ldots <p_k<}_{\text{small primes}}\underbrace{\underbrace{p_{k+1}<\ldots}_{\text{big primes}}}_{\displaystyle{\sum_{j>k}\dfrac{1}{p_j}<\dfrac{1}{2}}}$$
2. Set $N_b =\# \{n \leq N\,|\,\exists \,j>k\, : \,p_j|n\}$ and $N_s=\#\{n\leq N\,|\,p_j|n\Longrightarrow j\leq k\}.$
3. $N_b < \dfrac{N}{2}$
4. $N_s < 2^k\sqrt{N}$
5. $N_b+N_s < N$