Statistical quasi-Cauchy sequences
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Date
2011
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
PERGAMON-ELSEVIER SCIENCE LTD
Access Rights
info:eu-repo/semantics/openAccess
Abstract
A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (x(n)) statistically quasi-Cauchy if st - lim(n ->infinity) d(x(n+1), x(n)) = 0, and lacunary statistically quasi-Cauchy if S-theta - lim(n ->infinity) d(x(n+1), x(n)) = 0. We prove that a subset E of a metric space is totally bounded if and only if any sequence of points in E has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset E of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in E. (C) 2011 Elsevier Ltd. All rights reserved.
Description
Keywords
Quasi-Cauchy sequences, Slowly oscillating sequences, Summability, Total boundedness, Uniform continuity
Journal or Series
MATHEMATICAL AND COMPUTER MODELLING
WoS Q Value
Q1
Scopus Q Value
N/A
Volume
54
Issue
05.Jun
