This paper investigates discrete boundary value problems (BVPs) involving secondorder difference equations and two-point boundary conditions. General theorems guaranteeing the existence and uniqueness of solutions to the discrete BVP are established. The methods involve a sufficient growth condition to yield an a priori bound on solutions to a certain family of discrete BVPs. The a priori bounds on solutions to the discrete BVP do not depend on the step-size and thus there are no “spurious” solutions. It is shown that solutions of the discrete BVP will converge to solutions of ordinary differential equations.