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 $S=S_{nm}$ which almost converges to $L$, Lebesgue measure of the set of all its subsequences which almost converge to $L$ is either 1 or 0.