Summability of subsequences of a divergent sequence by regular matrices

Abstract

Stuart proved that the Cesaro matrix $C_1$ cannot sum almost every subsequence of a bounded divergent sequence $x$. 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 this note, we confirm Stuart's conjecture, and we extend it to the more general case of divergent sequences $x$.