We present two stability results in this paper. We first obtain sufficient conditions for the continuity of optimal values and solutions of convex programs in general vector spaces, as well as some types of robustness of some sub-optimal solutions. We then use these results in order to establish a new result in stochastic dynamic control of discrete event systems (known as constrained Markov Decision Processes): the convergence of the value and optimal policies of the problem with discounted costs, to the ones for the problem with expected average cost.