Sum of Reciprocals of Powers as Euler Product/Proof 1

Proof
From Euler Product:
 * $\displaystyle \sum_{n \mathop \ge 1} a_n n^{-z} = \prod_p \frac 1 {1 - a_p p^{-z} }$

$\displaystyle \sum_{n \mathop = 1}^\infty a_n n^{-z}$ is absolutely convergent.

For all $n \in \Z_{\ge 1}$, let $a_n = 1$.

From Sum of Reciprocals of Powers is Absolutely Convergent iff Modulus of Power is Greater than One:
 * $\displaystyle \sum_{n \mathop \ge 1} n^{-z}$ is absolutely convergent


 * $\left\lvert{z}\right\rvert \ge 1$
 * $\left\lvert{z}\right\rvert \ge 1$

Then it follows that:
 * $\displaystyle \sum_{n \mathop \ge 1} \frac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$