Exponential stability and asymptotic perturbed stability results are derived for nonlinear discrete-time systems. Some related results are available in the literature, but proofs for the discrete-time case are difficult to find. Our motivation for obtaining these results was to determine conditions under which discrete-time nonlinear model predictive control is stabilizing in the face of perturbations. This problem has particular relevance when output (rather than state) feedback is used. A nonlinear state estimator is then required to reconstruct the state, and it is necessary to identify the class of reconstruction errors for which the cascaded estimator/controller assembly is stabilizing.