A characterization of convergence for sequences (or series via their sequence of partial sums) in metric or uniform spaces: a sequence is Cauchy if its terms become arbitrarily close to each other as the index increases; in a complete space, Cauchy sequences converge.

Principle

Convergence without limits: the criterion separates the internal coherence of a sequence (mutual closeness of tails) from the existence of a limit point; completeness of the ambient space turns the internal Cauchy property into actual convergence.

Demonstration

The sequence x_n = 1/n in R is Cauchy because for large m,n the difference |1/n - 1/m| can be made arbitrarily small; the partial sums of the alternating harmonic series fail the Cauchy property and thus diverge in R.

Misapplication

Using the Cauchy criterion in a space that is not endowed with a uniform structure (pure topological spaces without a metric) without switching to nets or filters, or assuming a Cauchy sequence must converge regardless of completeness of the space.

Consequence

The criterion provides a limit-free test for convergence and is the defining tool for completeness; it underlies constructions of completions (e.g., completing the rationals to the reals) and is central in analysis and numerical convergence proofs.

Reversal

The reverse statement isolates spaces: existence of limits for all Cauchy sequences characterizes completeness. Conversely, a sequence can be Cauchy without converging precisely in incomplete spaces (e.g., rational Cauchy sequences without rational limits).

Boundary

Applies in metric and more generally uniform spaces; for general topological spaces one must use Cauchy nets/filters with respect to a uniformity. It does not directly handle convergence concepts that require order or measure structures without additional data.

Semantic Tension

Tension exists between the Cauchy property and pointwise convergence: Cauchy concerns pairwise tail closeness while pointwise or distributional convergence may allow oscillatory behavior; also between completeness (space property) and convergence (sequence property).

Synthesis

The Cauchy criterion isolates intrinsic convergence behavior by testing mutual closeness of sequence tails; combined with the ambient space's completeness it yields actual convergence, and where completeness fails it diagnoses the need for completion.