# 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:

$\size x < 1$ if and only if $\sequence {x_n}$ 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 if and only if $\sequence {x_n}$ is a null sequence in the real numbers
$\sequence {x_n}$ is a null sequence in the real numbers if and only if $\size x < 1$

$\blacksquare$