Summability-preserving functions

PROOF. Every subsequence of { Xn} converges, so that every subsequence of {f(xn) } is A-summable, by hypothesis. By [I], applied to complex sequences, {f(xn) } is actually convergent. (For the proof of [I] can be modified to apply to complex sequences. Or, see [2, Theorem 2], for a proof in a slightly more general context.) Let I xn } converge to x. The sequence xi, x, x2, x, * * * is also convergent, so that the sequence f(x1), f(x), f(x2), f(x), * * * converges. This shows that {f(xn) } converges to f(x) whenever { xn } converges to x. Thus the continuity of f is proved. The converse of the lemma is a consequence of the definition of Toeplitz matrix. The lemma is of course also true for real-valued functions of a real variable. The same comment applies also after the following theorem.