# 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="">
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## 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 an NSNLS [mathematical expression not reproducible] are Cauchy sequences of soft vectors and {[[??].sub.n]} is a Cauchy sequence of soft scalars in an NSNLS ([??](K), N, *, *), then [mathematical expression not reproducible] are also Cauchy sequences in NSNLS ([??](K),N, *, *).
Now, we will show that (g([x.sup.1.sub.k])),(g([x.sup.2.sub.k])),..., (g([x.sup.n.sub.k])) are Cauchy sequences.
For an integer m > 7/2, the solutions obtained in Proposition 8 form the Cauchy sequences in the following spaces:
As [L.sub.2](J,U), [L.sub.2](],V) are complete spaces, it is sufficient to prove {[u.sub.n,[lambda]]} and {[x.sub.n,[lambda]]} are Cauchy sequences in [L.sub.2](J, U), [L.sub.2](J, V), respectively
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.
On statistically convergent and statistically Cauchy sequences. Indian Journal of Pure and Applied Mathematics, v.
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.
As noted in , The elements of [[??].sup.p.sub.m] are technically equivalence classes of Cauchy sequences of elements of [[??].sub.0].
(The difference should not be exaggerated, for indeed the structuralist may welcome constructions based on relatively secure items to help establish coherence or realizability of the defining conditions.) 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.
We will show that {[z.sub.n]}, {[z'.sub.n]} in X are Cauchy sequences. Putting [mathematical expression not reproducible] in the equality (18), we get

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