In 1943 R. C. Buck gave a
summability characterization of real convergent sequences by showing that a real
sequence x is convergent if there exists a regular matrix summability method which
sums every subsequence of x. In 1944 R. P. Agnew generalized Buck’s result by
showing that if x is a bounded complex sequence and A is a regular matrix, then
there exists a subsequence y of x such that every limit point of x is a limit point of
Ay. In the present paper a theorem concerning “stretchings” of sequences is
proved; and from this theorem, summability characterizations of several
classes of sequences are obtained, together with an extension of Agnew’s
result.
