Editing Cauchy sequence (section)

===Examples===
The [[real number]]s are complete under the metric induced by the usual absolute value, and one of the standard [[Construction of the real numbers|constructions of the real numbers]] involves Cauchy sequences of [[rational number]]s. In this construction, each equivalence class of Cauchy sequences of rational numbers with a certain tail behavior—that is, each class of sequences that get arbitrarily close to one another— is a real number.

A rather different type of example is afforded by a metric space ''X'' which has the [[discrete space|discrete metric]] (where any two distinct  points are at distance 1 from each other). Any Cauchy sequence of elements of ''X'' must be constant beyond some fixed point, and converges to the eventually repeating term.