##  [Cauchy Sequence](/cauchy-sequence-0) 

 Definition

A sequence (x_n) in a metric space (X,d) is a Cauchy sequence if for every ε>0 there exists N such that for all m,n≥N one has d(x_m,x_n)<ε; terms become arbitrarily mutually close as the index grows.

 

 

 

 

 

 





## Principle

Principle

Tests intrinsic completeness: a metric space is complete if and only if every Cauchy sequence converges to a limit in the space, so Cauchy sequences detect missing limit points and motivate completions.

 

 

 

 

 





## Demonstration

Demonstration

In the real numbers with the usual metric, any Cauchy sequence converges to a real limit; by contrast, the rationals Q contain Cauchy sequences (e.g., approximations to √2) that do not converge in Q but do in R.

 

 

 

 

## Misapplication

Misapplication

Assuming a Cauchy sequence in a subspace automatically converges in that subspace without checking completeness (for example, treating a Cauchy sequence in Q as convergent in Q).

 

 

 

 

 





## Consequence

Consequence

Enables construction of completions (equivalence classes of Cauchy sequences), and underlies many convergence arguments in analysis and functional analysis by separating internal approximation from existence of limits.

 

 

 

 

## Reversal

Reversal

Sequences that have accumulation points but are not Cauchy (e.g., x_n = (−1)^n) do not have terms that become mutually arbitrarily close; boundedness does not imply Cauchy property.

 

 

 

 

 





## Boundary

Boundary

Definition depends on the metric (or uniform structure); in general topological spaces sequences may be insufficient to detect completeness and one uses Cauchy nets or filters in uniform spaces.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often conflated with convergence; in non-complete spaces Cauchy and convergent differ — conflating them obscures the need to check ambient completeness when asserting limits.

 

 

 

 

 





## Synthesis

Synthesis

A Cauchy sequence is one whose terms eventually become arbitrarily close to each other; it is the internal consistency notion that reveals whether the ambient space contains its limit points.