It is well known that if a
regular matrix sums every subsequence of a sequence x, then x converges. It follows
trivially from this result and row finiteness of the Cesáro summability matrix that if
A is a regular matrix such that Ay is Cesáro summable for every subsequence y of
x, then x is convergent (not merely Cesáro summable). The purpose of the present
paper is to give some general results of this type involving matrix methods that are
not necessarily row finite. For example, it is shown that if T is any regular matrix
summability method and A is a regular matrix such that Ay is absolutely
T-summable for every stretching y of x, then x is absolutely convergent. This is done
without assuming that x is bounded, and consequently, without the benefit of
associativity.
