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 Rational Sequence Converges in Rationals iff Converges in Reals then:
 * $\sequence {x_n}$ 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 then:
 * $\sequence {x_n}$ is a null sequence in the Real numbers $\size{x} < 1$