On a theorem of W. Meyer-König and H. Tietz

Let (un) be a sequence of real numbers and let L be an additive limitable method with some property. We prove that if the classical control modulo of the oscillatory behavior of (un) belonging to some class of sequences is a Tauberian condition for L, then convergence or subsequential convergence of (un) out of L is recovered depending on the conditions on the general control modulo of the oscillatory behavior of different order.