The Wikipedia page on the divergence of the sum of reciprocals of primes has a proof of the log log divergence rate.
It starts with the following inequality:
$$\sum_i^n \frac{1}{i} \leq \prod_{p\leq n}(1+\frac{1}{p}) \cdot \sum_{k=1}^n \frac{1}{k^2}$$
The inequality is stated as obvious but it isn't to me and those I've asked.
Can anyone explain the inequality?