Almost-convergence of double sequences (subsequences) is equivalent to almost Cauchy condition. If the set of all almost convergent subsequences of a sequence S = Snm 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. .
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