Let ( M , d ) be a finite-dimensional complete metric space, and { T n } a sequence of uniformly convergent operators on M . We study the non-autonomous discrete dynamical system x n + 1 = T n x n and the globally asymptotic stability of the inhomogeneous iterates of { T n } . Then we apply the results to investigate the stability of equilibrium of T when it satisfies certain type of sublinear conditions with respect to the partial order defined by a closed convex cone. The examples of application to nonlinear difference equations are also given.