On matrix summability of spliced sequences and A-density of points

For y∈Ry∈R and a sequence x=(xn)∈ℓ∞x=(xn)∈ℓ∞ we define the new notion of A  -density δA(y)δA(y) of indices of those xnxn's which are close to y where A is a non-negative regular matrix. We present connections between A  -densities δA(y)δA(y) of indices of (xn)(xn) and the A  -limit of (xn)(xn). Our main result states that if the set of limit points of (xn)(xn) is countable and δA(y)δA(y) exists for any y∈Ry∈R where A   is a non-negative regular matrix, then limn→∞⁡(Ax)n=∑y∈RδA(y)⋅y, which presents a different view of Osikiewicz Theorem.