Sequence of Powers of Number less than One/Necessary Condition

Theorem
Let $x \in \R$ be such that $\size{x} < 1$.

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

Then $\sequence {x_n}$ is a null sequence.

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