# Sequence of Powers of Number less than One/Rational Numbers

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## 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

By Sequence of Powers of Real Number less than One:

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

$\blacksquare$