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In a metric space, every convergent sequence is Cauchy
Convergent sequence is Cauchy: Let be a metric space and let be a convergent sequence with in . Then is a Cauchy sequence; that is, for every there exists such that if then .
Together with the converse implication in complete metric spaces, this links limits to intrinsic “eventual closeness” of sequence terms.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
A convergent sequence in a topological space converges to a point if for every neighborhood of , there exists such that for all .
In a metric space , this is equivalent to . In a Hausdorff space, limits of convergent sequences are unique (see uniqueness of limits).
A Cauchy sequence in a metric space is a sequence such that for every there exists with
A complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .