  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.
