Cauchy sequence

A sequence in a metric space is Cauchy if, for every positive distance, all sufficiently late terms lie within that distance of one another. Its terms need not converge within the space.

Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. The metric supplies the distances used to define a Cauchy sequence.
Sequence: An ordered list of terms indexed by the natural numbers or another index set. Cauchy behavior concerns the distances among terms as their indices increase.
Harmonic series: The infinite series whose terms are the reciprocals of the positive integers; its partial sums diverge. Its partial sums are not Cauchy because tails can remain large.
Augustin-Louis Cauchy: A French mathematician whose work helped establish rigorous foundations for analysis. The sequence condition bears his name and reflects his influence on analysis.
Cauchy criterion: A condition that a sequence's terms become uniformly close to one another after some index. It expresses the defining test directly in terms of pairwise distances.
Metric: A function assigning distances to pairs of points and satisfying the metric axioms. A metric turns the phrase “close to one another” into a precise quantity.
Geometric series: A series whose terms form a geometric progression with a constant ratio. When the ratio has absolute value below one, its partial sums form a Cauchy sequence.
Cauchy's criterion for convergence: A test characterizing convergence through the behavior of sequence tails or series remainders. Cauchy developed criteria that let convergence be studied without first knowing a limit.
Convergent sequence: A sequence that approaches a limit in its space. Every convergent sequence is Cauchy, but the converse depends on the space.
Epsilon–N definition: A formal limit definition using an arbitrary positive tolerance and a sufficiently large index. The Cauchy definition uses the same quantifier pattern, applied to pairs of terms.

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