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

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

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

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

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

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

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

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

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.