# 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 ?
thus I ([u.sub.n],[v.sub.n]) - 0, i.e I ([u.sub.n],[v.sub.n]) is a Cauchy sequence in X*.
Then, there is a study on Cauchy sequence in an NSNLS in Section 4.
The Cauchy sequence has the following two important properties [23, 24]:
Then a sequence {[x.sub.n]} in X is called a Cauchy sequence if for every [epsilon] > 0, there exists K([epsilon]) [member of] N, such that d([x.sub.n], [x.sub.m]) < [epsilon] for all n, m [greater than or equal to] K([epsilon]).
Hence, {[x.sub.n]} is a Cauchy sequence and then converges to [x.sub.[omega]] [member of] X.
(3) (X,d) is complete if and only if every Cauchy sequence in X is convergent to some point in X.
(2) The sequence {[x.sub.n]} is said to be a Cauchy sequence in (X, [[sigma].sub.b]) if [lim.sub.n[right arrow][infinity]] [[sigma].sub.b]([x.sub.n], [x.sub.m]) exists and is finite;
Now it is to prove that {[y.sub.n]} is a Cauchy sequence. Suppose that d([y.sub.n], [y.sub.n+1]) = 0 for some n > 0.
It is not difficult to show that {[U.sup.(n)]} with the initial data ([u.sup.(n)], [v.sup.(n)], [q.sup.(n)]) = ([u.sub.0], [v.sub.0], [q.sub.0]) is the Cauchy sequence in [C.sup.0](J; [H.sup.2]), which proves the following theorem.
Second, we show that, for some finite time T, the sequence [{[[PI].sub.[epsilon]]}.sub.[epsilon]>0] is a Cauchy sequence in C(0, T; [L.sup.2]).
(Step 4) Formulas [(2).sub.m] of Step 3 imply that (22) defines a Cauchy sequence [{[e.sup.(m)]}.sub.m[greater than or equal to]0] in the Banach space [mathematical expression not reproducible].

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