Sequence of Powers of Number less than One/Rational Numbers

Theorem
Let $x \in \Q$.

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

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

Proof
By the definition of convergence of a rational sequence:
 * $\sequence {x_n}$ is a null sequence in the rational numbers $\sequence {x_n}$ is a null sequence in the real numbers

By Sequence of Powers of Real Number less than One:
 * $\sequence {x_n}$ is a null sequence in the real numbers $\size x < 1$