Sequence of Powers of Number less than One

Theorem
Let $x \in \R$.

Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$.

Then:
 * $\size x < 1$ $\sequence {x_n}$ is a null sequence.

Necessary Condition
[For other proofs of the Necessary Condition visit here.]