TITLE: Why can infinite product only be zero if one of the factors is zero?
QUESTION [5 upvotes]: I was reading about the Riemann zeta function in the region $Re(Z) > 1$, where
it can be represented by the Euler product formula. And the book mentioned that
there can be no zeros in this region, since each factor  $1/(1 - 1/p^s)$ is never zero.
Can someone explain why the product can't converge to zero? For example, 
consider the infinite product $1/2 \cdot 1/3 \cdot 1/4 \cdots$.  This seems to converge to zero,
but each factor is not zero.
Thanks!

REPLY [2 votes]: Here they explain your question.
In particular they show 
$$\prod_{k=1}^{\infty} \frac{1}{1-\frac{1}{p_k}}=\sum_{n=1}^{\infty} \frac{1}{n^s}>1$$