A.: Summability of certain category two classes

ABSTRACT. The main purpose of this paper is to show that a sequence x is necessarily convergent if there exists a non-Schur matrix A with convergent columns such that the set of A-summable subsequences (rearrangements) of x is of the second category. The key to obtaining these results lies in the following theorem due to |. J. Maddox and an analog to it in which "subsequence " is replaced with "rearrangement. " The matrix A is Schur if there exists a divergent sequence x such that each subsequence of x is A-summable. Let x be a fixed sequence. For each t G (0,1], we write t in its nonterminating binary decimal expansion and pair it with the subsequence y of x in which x k is retained if and only if there is a 1 in the koth decimal place in the expansion of t. It follows that K = { subsequences of x} is of the second category in itself. In addition, by utilizing the natural map of J = {rearrangements of x} onto ß = { rearrangements of I +} and the distance functio