We consider a summability matrix A with sectional convergence (in the null-domain) and use sectional submethods B. If (IB)s has small oscillation (for a suitable B), then s also has small oscillation. Employing a set of such B, we obtain Tauberian results yielding convergence (also with one-sided conditions). In particular this analysis applies to weighted means. Thus we have a new approach to results given by Móricz and Rhoades .