Hi everybody, I have this tricky inequality

$\displaystyle |\zeta (z) \prod_{k=1}^{N}(1-\frac{1}{p_k^z})-1|\leq \sum_{n=p_{N+1}}^\infty \frac{1}{n^{Re z}}, $ Re z > 1
where $\displaystyle \zeta(z)$ is the Riemann zeta function $\displaystyle \sum_{n=1}^\infty \frac{1}{n^z}$, Re z > 1 and $\displaystyle p_k$ is the sequence of prime numbers.

I seem to have found something useful on the topic here (page 2)
An Introduction to the Theory of the ... - Google Bogsøgning

But they're doing it, in my opinion, in a quite difficult way, something about induction over the prime numbers in another identity and then the inequality follows as a result, and my math skills aren't quite on that level . So I was just wondering if there was some easier way of doing this by, for example using that the product gets smaller and smaller the bigger the number N gets?