In a paper concerning a quick calculation of Bernoulli numbers the following inequality is presented (page 3), only referring to it as "not hard to see" that it holds.
$$ \sum_{n \leq N}^{} n^{-s} \leq \prod_{p \leq N}^{} (1 - p^{-s})^{-1}$$
where $p \in \mathbb{P}$.
I believe it is not too hard to see, but I guess I miss some elementary background. Could someone explain how to derive this inequality?
Hint. Try $N=4$ and expand and multiply the two geometric series on the right.