Definition:Cauchy Sequence/Rational Numbers

Definition

Let $\sequence {x_n}$ be a rational sequence.

Then $\sequence {x_n}$ is a Cauchy sequence if and only if:

$\forall \epsilon \in \Q_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {x_n - x_m} < \epsilon$

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.

Source of Name

This entry was named for Augustin Louis Cauchy.