Sum of Infinite Geometric Sequence/Proof 1

Proof
From Sum of Geometric Progression, we have:
 * $\displaystyle s_N = \sum_{n \mathop = 0}^N z^n = \frac {1 - z^{N+1}} {1 - z}$

We have that $\left \vert {z}\right \vert < 1$.

So by Sequence of Powers of Number less than One:
 * $z^{N+1} \to 0$ as $N \to \infty$

Hence $s_N \to \dfrac 1 {1 - z}$ as $N \to \infty$.

The result follows.

It remains to demonstrate absolute convergence:

The absolute value of $\left \vert {z}\right \vert$ is just $\left \vert {z}\right \vert$.

By assumption:
 * $\left \vert {z}\right \vert < 1$

So $\left \vert {z}\right \vert$ fulfils the same condition for convergence as $z$.

Hence:
 * $\displaystyle \sum_{n \mathop = 0}^\infty \left \vert {z}\right \vert^n = \frac 1 {1 - \left \vert {z}\right \vert}$