Cauchy criterion

The Cauchy criterion states that a sequence in a complete metric space converges if and only if its terms eventually become arbitrarily close to one another.

Cauchy sequence: A sequence whose terms eventually lie arbitrarily close to one another. The criterion is expressed precisely by this property of a sequence.
Sequence: An ordered list of objects indexed by natural numbers or another index set. The criterion applies to the terms of such an indexed list.
Cauchy condensation test: A convergence test relating a decreasing positive series to the series formed from its terms at powers of two. It bears Cauchy's name and uses a distinct test for series convergence.
Monotone convergence theorem: A theorem stating that every bounded monotone real sequence converges. It proves convergence using order and boundedness rather than pairwise closeness.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. Distances between sequence terms give the criterion its meaning.
Convergence: Approach toward a limit or stable value under a specified notion of distance or topology. The criterion characterizes convergence through an internal condition on terms.
Uniform convergence: Convergence of functions in which one index bound works uniformly across the entire domain. Uniform Cauchy criteria test function sequences by pairwise differences across all inputs.
Bolzano–Weierstrass theorem: A theorem stating that every bounded sequence in Euclidean space has a convergent subsequence. It guarantees a convergent subsequence, not convergence of the whole sequence.
Completeness (mathematics): A property of a metric space in which every Cauchy sequence converges within the space. Completeness is the condition that makes the criterion imply convergence.
Real number: An element of the complete ordered field that represents points on the number line. For real sequences, the criterion is equivalent to convergence in the real numbers.

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