Sum of Infinite Geometric Sequence/Corollary 1/Proof 2

Corollary to Sum of Infinite Geometric Progression
Let $S$ be a standard number field, i.e. $\Q$, $\R$ or $\C$.

Let $z \in S$.

Let $\left \vert {z}\right \vert < 1$, where $\left \vert {z}\right \vert$ denotes:
 * the absolute value of $z$, for real and rational $z$
 * the complex modulus of $z$ for complex $z$.

Then:


 * $\displaystyle \sum_{n \mathop = 1}^\infty z^n = \frac z {1 - z}$