Hi everybody, I have this tricky inequality

Re z > 1

where is the Riemann zeta function , Re z > 1 and 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?