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 ?
It is defined also the concept of f--statistically Cauchy sequence and it is proved that if X is a complete space and [([x.
iii) If every Cauchy sequence in X is convergent, then (X,d) is said to be a complete complex valued b-metric space.
A multiplicative metric space (X, m) is complete if every multiplicative Cauchy sequence in it is multiplicative convergent to some x E X, [16].
A normed vector space V is called a Banach space if every Cauchy sequence [{[w.
is a Cauchy sequence in the Banach space C(I,R), hence converging uniformly to some x(.
Finally, S [subset] X is forward complete if every forward Cauchy sequence is forward convergent.
is said to be strongly complete iff every strong Cauchy sequence in V is strongly convergent to a point in V.
j=1] is a Cauchy sequence of complex numbers for each fixed k [member of] [N.
By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.
Song [14] has pointed out there are some errors in the papers of Vasuki [15] and Grabiec [4], because definition of Cauchy sequence given by Grabiec [4] is weaker than the one proposed by Song [14] and hence conditions of Vasuki's theorem and its corollary.