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 $\size x < 1$
$\blacksquare$