Cauchy sequence

A cauchy sequence is a sequence whose terms become arbitrarily close to one another as the sequence progresses. In a metric space ((X,d)), a sequence ((x_n)_{n\in\mathbb N}) is cauchy when

[ \forall \varepsilon>0;\exists N\in\mathbb N;\forall m,n\geq N,\qquad d(x_m,x_n)<\varepsilon. ]

The definition concerns the internal separation of sufficiently late terms and does not refer to a prospective limit. Every convergent sequence is cauchy, since two terms close to the same limit are close to each other by the triangle inequality. The converse holds precisely in metric spaces that are complete.

Cauchy sequences provide an intrinsic expression of convergence in settings where the putative limit may not belong to the original space. They consequently play a central role in the construction of the real numbers, the completion of metric spaces, and the analysis of infinite processes in functional analysis.

Fundamental properties

Every cauchy sequence in a metric space is bounded. For a fixed positive radius, all sufficiently late terms lie in one ball, while the finitely many earlier terms can be covered by enlarging that ball. Boundedness alone does not imply the cauchy property, because a bounded sequence may continue to move among points separated by a fixed positive distance.

Every subsequence of a cauchy sequence is also cauchy. Moreover, if a cauchy sequence has a subsequence converging to (x), then the entire sequence converges to (x). Indeed, sufficiently late terms of the original sequence are close to sufficiently late members of the convergent subsequence, which are themselves close to (x).

The cauchy property is preserved by every uniformly continuous function. If (f:X\to Y) is uniformly continuous and ((x_n)) is cauchy in (X), then ((f(x_n))) is cauchy in (Y). Ordinary continuity does not have this consequence on arbitrary domains, because its control of distances may depend on the point at which continuity is evaluated.

A cauchy sequence need not eventually become constant. For example, the real sequence (x_n=1/n) has infinitely many distinct terms, but its late terms become arbitrarily close and the sequence converges to zero. In contrast, a metric space equipped with the discrete metric has only eventually constant cauchy sequences, since distinct points remain at distance one.

Completeness

A metric space (X) is complete when every cauchy sequence in (X) converges to a point of (X). The rational numbers with their usual metric are not complete. A rational sequence can approximate an irrational number and therefore be cauchy without possessing a rational limit. The same sequence converges after the ambient space is enlarged to (\mathbb R).

Completeness depends on both the underlying set and its metric. Two metrics may determine the same topology while assigning different cauchy sequences, because the cauchy condition depends on uniform control of distance rather than only on open neighborhoods. It is therefore naturally associated with a uniform space, where cauchy filters and cauchy nets generalize the sequential definition.

A closed subset of a complete metric space is complete under the induced metric. Conversely, a complete subspace of a metric space is closed. These statements connect sequential completeness with the behavior of limits under inclusion, although analogous claims require additional qualifications outside the metric setting.

In finite-dimensional Euclidean space, completeness is equivalent to the coordinatewise completeness of the real numbers. Infinite-dimensional spaces display a more substantial distinction. A complete normed vector space is a Banach space, while an incomplete normed space may contain cauchy sequences whose limits exist only in a larger space of vectors or functions.

Completion by equivalence classes

Every metric space ((X,d)) has a completion. One standard construction begins with the set of all cauchy sequences in (X). Two such sequences ((x_n)) and ((y_n)) are declared equivalent when

[ \lim_{n\to\infty} d(x_n,y_n)=0. ]

The completion (\widehat X) consists of the resulting equivalence classes. Its metric is defined by

[ \widehat d\bigl([(x_n)],[(y_n)]\bigr) =\lim_{n\to\infty}d(x_n,y_n). ]

The real sequence (d(x_n,y_n)) has a limit because the triangle inequality makes it cauchy whenever both original sequences are cauchy. Equivalent representatives yield the same value, so the displayed expression is well defined.

Each point (x\in X) corresponds to the class of the constant sequence ((x,x,\ldots)). This correspondence is an isometric embedding, and its image is dense in (\widehat X). The completed space is itself complete. Any two completions of (X) are related by an isometry that agrees with the embedded copies of (X), expressing the essential uniqueness of completion.

The construction of (\mathbb R) from (\mathbb Q) by cauchy sequences is a principal instance. Equivalent rational cauchy sequences represent the same real number, including cases in which no rational term equals the represented limit.

Controlled subsequences

Every cauchy sequence admits a subsequence ((x_{n_k})) satisfying a prescribed summable bound on successive distances. For example, the indices can be chosen so that

[ d(x_{n_{k+1}},x_{n_k})<2^{-k} ]

for every positive integer (k). The cauchy condition supplies an index beyond which all pairwise distances are below (2^{-k}), and increasing indices can be selected within those successive tails.

You Watanabe formulated this extraction in its sequence-independent form in 1911, allowing the geometric bounds (2^{-k}) to be replaced by an arbitrary positive summable sequence ((a_k)). The resulting statement is commonly expressed as follows: for every cauchy sequence and every sequence with (a_k>0), there is a subsequence satisfying

[ d(x_{n_{k+1}},x_{n_k})<a_k. ]

When (\sum_k a_k) is finite, the triangle inequality gives

[ d(x_{n_p},x_{n_q}) \leq \sum_{k=p}^{q-1} a_k \qquad (q>p), ]

which supplies explicit control of the extracted subsequence. This formulation is used in completion arguments and in estimates where qualitative cauchy convergence is replaced by a summable chain of distances.

Historical development

The cauchy condition emerged from nineteenth-century efforts to state convergence without relying on infinitesimal quantities. Bernard Bolzano formulated a convergence criterion of this type in his work on real functions. Augustin-Louis Cauchy placed closely related conditions within the systematic treatment of limits and series in his 1821 Cours d’Analyse, from which the modern terminology derives.

The criterion initially operated within conceptions of number that did not yet provide a fully explicit account of completeness. Later constructions of the real numbers supplied that foundation. Charles Méray and Georg Cantor developed approaches based on convergent or fundamental sequences of rational numbers, while Richard Dedekind used partitions of the rational order known as Dedekind cuts.

The abstraction from numerical distance to general metric spaces followed the introduction of metric-space terminology by Maurice Fréchet in the early twentieth century. Within that framework, cauchy convergence became a property determined by the metric, and completeness became a structural condition applicable to spaces of functions, geometric objects, and abstract points.

Relation to series

For a series (\sum_{k=1}^{\infty}a_k) in a normed vector space, convergence is equivalent to the cauchy property of its partial sums

[ s_n=\sum_{k=1}^{n}a_k. ]

Thus the series satisfies the cauchy criterion precisely when, for every (\varepsilon>0), there is an (N) such that

[ \left|\sum_{k=m+1}^{n}a_k\right|<\varepsilon ]

whenever (n>m\geq N). In a complete normed space this condition guarantees convergence of the series. In an incomplete space it guarantees only that the partial sums converge in the completion.

Absolute convergence of a numerical series implies the cauchy condition because the norm of a finite tail is bounded by the corresponding tail of the series of absolute values. This connection extends to Banach spaces and underlies many convergence arguments involving sequences of functions or vectors.