This note proposes a bounded nonlinear control law composed of saturation functions for the discrete-time chain of integrators. A dynamical adaptation rule of the saturation levels involved in the control law is proposed to improve the closed-loop performances. The note unifies the original work of Yang et al. (1997) with static saturation level and convergence improvements that recently appeared in the continuous time case. The possible ranges for the controller's parameters are extended with respect to existing results.