Summability of Subsequences of a Divergent Sequence by Regular Matrices II

Abstract

C. Stuart proved in [27, Proposition 7] that the Cesaro matrix C1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked ‘It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this’. In [4, Theorem 3.1] Stuart’s conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note we show that [4, Theorem 3.1] is a special case of Theorem 3.5.5 in [24] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.