In this note, continuing in the line of [2] we further consider a more general approach and for y ∈ R and a sequence x = (x n) ∈ ∞ we define the more general notion of IA-density of indices of those x n 's which are close to y, denoted by Iδ A (y) where A is a non-negative regular matrix. Connections are drawn between Iδ A (y) and particular limit points of ((Ax) n). Our main result states that if x = (x n) is a bounded sequence, Iδ A (y) exists for every y ∈ R and y∈D Iδ A (y) = 1 then I − lim n→∞ (Ax) n = y∈D Iδ A (y) · y provided both finitely exists. This is an improvement of the alternative version of famous Osikiewicz Theorem given in [2].