Sequence of Powers of Number less than One/Complex Numbers

Theorem
Let $z \in \C$.

Let $\sequence {z_n}$ be the sequence in $\C$ defined as $z_n = z^n$.

Then:
 * $\size{z} < 1$ $\sequence {z_n}$ is a null sequence.

Proof
By the definition of convergence then:
 * $\displaystyle \lim_{n \to \infty} z_n = 0 \iff \lim_{n \to \infty} \size {z_n} = 0$

By Modulus of Product then for each $n \in \N$:
 * $\size {z_n} = \size {z^n} = \size {z}^n$.

So:
 * $\displaystyle \lim_{n \to \infty} \size {z_n} = 0 \iff \lim_{n \to \infty} \size {z}^n = 0$

Since $\size {z} \in \R_{\ge 0}$, by Sequence of Powers of Real Number less than One then:
 * $\displaystyle \lim_{n \to \infty} \size {z}^n = 0 \iff \size {z} < 1$

The result follows.

Also known as
This result and Power of Reciprocal are sometimes referred to as the basic null sequences.