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Uniform statistical convergence of double subsequences
Authors: Fikret Čunjalo
Number of views: 525
In the [3] is proven that sequence Sij uniformly statistically converges to L if and only if it there is a subset A of the set N N uniform density zero and subsequence S (x) dened by, Sij (x) = Sij for (i; j) 2 Ac, converges to L, in the Pringsheim's sense. In this paper it is proven that analog is valid for subsequence S (x) provided that for each N and i 6 N _ j 6 N is a set of all Sij (x) nite set. Is generally valid: If the subsequence S (x) uniformly statistically converges to L, then, there is a subset A of the set N N uniform density zero and subsequence S (y) dened by, Sij (y) = Sij (x) for (i; j) 2 Ac, converges to L, in the Pringsheim's sense. If there is a subset
A of the set N N uniform density zero and subsequence S (y) dened by, Sij (y) = Sij (x) for (i; j) 2 Ac, such that limi!1( limj!1Sij (y)) = L, then, the
subsequence S (x) uniformly statistically converges to L.