Sum of Reciprocals of Powers as Euler Product/Proof 1

Proof
By definition of Euler product:
 * $\ds \sum_{n \mathop = 1}^\infty a_n n^{-z} = \prod_p \frac 1 {1 - a_p p^{-z} }$

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

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

From Convergence of P-Series:
 * $\ds \sum_{n \mathop = 1}^\infty n^{-z}$ is absolutely convergent


 * $\cmod z > 1$
 * $\cmod z > 1$

It then follows that:
 * $\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z} }$