Number of Primes is Infinite/Proof 4

Proof
there exist only a finite number of primes.

From Sum of Reciprocals of Powers as Euler Product:
 * $\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 {n^z} = \prod_p \frac 1 {1 - p^{-z}}$

When $z = 1$ this gives:
 * $\displaystyle \sum_{n \mathop \ge 1} \dfrac 1 n = \prod_p \frac 1 {1 - 1/p}$

As by hypothesis there exist only a finite number of primes, the is also finite.

But from Sum of Reciprocals is Divergent, the diverges to infinity.

The result follows by Proof by Contradiction.