# 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 ?
n]} is Cauchy sequence in X provided that sq < 1.
n]} is a Cauchy sequence with respect to intuitionistic fuzzy metric [M.
is called complete if every Cauchy sequence convergent with respect to [[tau].
iii) A Menger space in which every Cauchy sequence is convergent is said to be complete.
n]} is Cauchy sequence and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some x* [member of] C.
n] in V is called strongly Cauchy sequence if for every [lambda] > 0, there exists a positive integer N such that [v.
A normed vector space V is called a Banach space if every Cauchy sequence [{[w.
Finally, S [subset] X is forward complete if every forward Cauchy sequence is forward convergent.
n] [member of] X, then the sequence of partial sums forms a Cauchy sequence in X.
For more generality, we can replace the condition of closedness of T by the following condition: T is a nonempty subset of R such that every Cauchy sequence in T converges to a point of T with the possible exception of Cauchy sequences which converge to a finite infimum or finite supremum of t.
By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

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