 The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.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 metric (where any two distinct points are at distance 1 from each other).