Uniform limit of discrete convex functions

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform limit on compact subsets of discrete convex mesh functions which are uniformly bounded is a continuous convex function. Furthermore, if the discrete convex mesh functions interpolate boundary data of a continuous convex function and their Monge-Ampere masses are uniformly bounded, the limit function satisfies the boundary condition strongly. The domain of the solution needs not be uniformly convex. The result is applied to the convergence of some numerical methods for the Monge-Ampere equation.