Subsequential Convergence Conditions

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 (V (0) n (Δu)) generated regularly by a sequence in a class of sequences is a subsequential convergence condition for L, then for any integer m ≥ 1, the mth repeated arithmetic means of (V (0) n (Δu)), (V (m) n (Δu)), generated regularly by a sequence in the class (m), is also a subsequential convergence condition for L.