We demonstrate the existence of a smooth control-Lyapunov function (CLF) for difference equations asymptotically controllable to closed sets. We further show that this CLF may be used to construct a robust feedback stabilizer. The existence of such a CLF is a consequence of a more general result on the existence of a weak Lyapunov function under the assumption of weak asymptotic stability of a closed (not necessarily compact) set for a difference inclusion.