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:

$\size z < 1$ if and only if $\sequence {z_n}$ is a null sequence.

Proof

By the definition of convergence:

$\ds \lim_{n \mathop \to \infty} z_n = 0 \iff \lim_{n \mathop \to \infty} \size {z_n} = 0$

By Modulus of Product:

$\forall n \in \N: \size {z_n} = \size {z^n} = \size z^n$

So:

$\ds \lim_{n \mathop \to \infty} \size {z_n} = 0 \iff \lim_{n \mathop \to \infty} \size z^n = 0$

Since $\size z \in \R_{\ge 0}$, by Sequence of Powers of Real Number less than One:

$\ds \lim_{n \mathop \to \infty} \size z^n = 0 \iff \size z < 1$

The result follows.

$\blacksquare$

Also known as

This result and Sequence of Powers of Reciprocals is Null Sequence are sometimes referred to as the basic null sequences.

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Example $\text {(iv)}$