\begin{definition}[Definition:Cauchy Sequence/Rational Numbers]
Let $\left \langle {x_n} \right \rangle$ be a rational sequence.
Then $\left \langle {x_n} \right \rangle$ is a '''Cauchy sequence''' {{iff}}:
: $\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{x_n - x_m}\right| < \epsilon$
where $\Q_{>0}$ denotes the set of all strictly positive rational numbers.
Considering the set of rational numbers as a metric space, it is clear that this is a special case of the definition for a metric space.
{{namedfor|Augustin Louis Cauchy}}
Category:Definitions/Analysis
Category:Definitions/Sequences
\end{definition}