Subsequential Convergence Conditions

Abstract

Let (un) be a sequence of real numbers and let L be any (C,1) regular limitable method. We prove that, under some assumptions, if a sequence (un) or its generator sequence (Vn(0)(Δu)) generated regularly by a sequence in a class 𝒜 of sequences is a subsequential convergence condition for L, then for any integer m≥1, the mth repeated arithmetic means of (Vn(0)(Δu)), (Vn(m)(Δu)), generated regularly by a sequence in the class 𝒜(m), is also a subsequential convergence condition for L.