Almost-convergence of double sequences (subsequences) is equivalent to almost Cauchy condition. If the set of all almost convergent subsequences of a sequence S = S nm is of the second category, then S is convergent in the simple sense. For the sequence S = Snm which almost converges to L, Lebesgue measure of the set of all its subsequences which almost converge to L is either 1 or 0. 1 pq p−1 i=0 q−1 j=0 S n+i,m+j − L < ε for ∀p, q > N and ∀(n, m) ∈ N × N. We write f − lim S = L. We denote by X the set of all double sequences of 0's and 1's,namely, X = {x = (x nm) : x nm ∈ {0, 1}, n, m ∈ N} In [4] F. Moricz dened the concept of a subsequence of a double sequence.
Conditions are given on a nonnegative regular summability matrix A to ensure that for a given number α, 0 5 α 5 1, there exists a sequence x consisting of 0's and 1's such that Ax converges to α.
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