# Cauchy sequence

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## Cauchy sequence

[kō·shē ′sē·kwəns]
(mathematics)
A sequence with the property that the difference between any two terms is arbitrarily small provided they are both sufficiently far out in the sequence; more precisely stated: a sequence {an } such that for every ε > 0 there is an integer N with the property that, if n and m are both greater than N, then | an-am | < ε.="" also="" known="" as="" fundamental="" sequence;="" regular="">

## Cauchy sequence

(mathematics)
A sequence of elements from some vector space that converge and stay arbitrarily close to each other (using the norm definied for the space).
References in periodicals archive ?
In Section 2 we study some properties of f--statistical convergent and f--statistical Cauchy sequences, giving a characterization of Banach spaces in those terms.
2 On f--statistical convergence and f--statistical Cauchy sequences
In this section we try to make clear the role of Cauchy sequences in the setting of f--statistical convergence.
Also we find out the relation between [lambda]-statistical convergent and [lambda]-statistical Cauchy sequences in this spaces.
122) that there exist Cauchy sequences which are not Mackey-Cauchy sequences.
m] are technically equivalence classes of Cauchy sequences of elements of [[?
The constructivist takes real numbers as, say, Cauchy sequences of rationals (either equivalence classes thereof or certain canonical ones), but in the definition of 'Cauchy sequence', constructive quantifiers are to be understood.
One then speaks of 'constructive Cauchy sequences'; if Church's Thesis (CT) is accepted, these become identified with recursive Cauchy sequences, of which there are only countably many.
Given real numbers, x and y, generated by Cauchy sequences of rationals, <[x.
1,l]) (k,l [less than or equal to] r) are Cauchy sequences in (X,q).

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