Cauchy sequence

A Cauchy sequence is one whose terms eventually lie arbitrarily close to each other. In a metric space with distance function d, a sequence (x_n) is Cauchy if, for every ε > 0, there is an index N such that d(x_m, x_n) < ε whenever m and n are both at least N. The condition compares pairs of late terms rather than comparing terms with a proposed limiting point.

Intuitively, the sequence becomes self-contained inside successively smaller neighborhoods. A convergent sequence is always Cauchy because points sufficiently near the same limit are near one another; the converse depends on the space in which the sequence lives.¹ The definition applies to real, complex, normed, and more general metric spaces.

A Cauchy sequence converges precisely when the ambient space has enough points to contain all of its would-be limits. A metric space is called complete when every Cauchy sequence converges to a point of that space. The real numbers are complete, whereas the rational numbers are not: rational decimal approximations to the square root of 2 form a Cauchy sequence in the rational metric, but their limit is not rational.

Completeness is a property of both a set and its metric, not merely of the sequence. The rational sequence above converges in the real numbers but fails to converge within the rationals. The completion of a metric space enlarges it minimally, in a canonical sense, so that missing limits are added; the real numbers can be constructed as equivalence classes of Cauchy sequences of rationals.

Standard examples show how the definition operates without requiring an explicit limit. Every eventually constant sequence is Cauchy, and every sequence in a finite metric space that is Cauchy is eventually constant. In the real numbers, the sequence defined by x_n = 1/n is Cauchy because late terms are all close to zero, while the sequence x_n = (−1)ⁿ is not Cauchy because its even and odd subsequences remain a fixed distance apart.

A practical criterion in a normed vector space is the summable-increments test: if the series of norms of successive differences, Σ ||x_n+1 − x_n||, converges, then the partial sums form a Cauchy sequence. This estimate underlies the Banach fixed-point theorem, where repeated iteration is shown to be Cauchy and completeness supplies the fixed point.²

Cauchy sequences can encode points that are absent from the original space, which makes them a construction tool rather than only a convergence test. Two Cauchy sequences can be declared equivalent when their term-by-term distance tends to zero; the equivalence classes provide the points of a completion, with distance defined independently of the chosen representatives.

The idea also extends beyond ordinary numerical distance. In a uniform space, entourages replace a single metric and express when two points are uniformly close; Cauchy filters and Cauchy nets generalize sequence-based convergence in spaces where sequences do not capture all topological behavior. In computation, stopping rules based on the distance between successive approximations resemble the Cauchy condition, but a numerical tolerance alone does not prove convergence unless an error bound or completeness argument is available.

Metric space

A set equipped with a distance function satisfying nonnegativity, symmetry, the triangle inequality, and separation of distinct points.

Complete metric space

A metric space in which every Cauchy sequence converges to a point of the space.

Completion

A complete space containing the original metric space densely, usually obtained by adjoining limits represented by Cauchy sequences.

Uniform space

A structure that formalizes uniform notions of closeness and convergence without requiring a particular metric.