A sequence a_1, a_2, ... such that the metric d(a_m, a_n) satisfies lim_(min(m, n)->∞) d(a_m, a_n) = 0. Cauchy sequences in the rationals do not necessarily converge, but they do converge in the reals. Real numbers can be defined using either Dedekind cuts or Cauchy sequences.

Dedekind cut

Augustin-Louis Cauchy