In this paper, a continuous-time optimal control problem is approached in a sub-optimal way by introducing the concept of suboptimal value function, which is any function satisfying the Hamilton-Jacobi-Bellman inequality. It is shown that as long as the Euler Approximating System (EAS) of a given continuous-time plant admits a positive definite convex suboptimal value function, it is possible to determine a stabilizing control for the continuous-time system whose cost not only converges to the optimal, but it is also upper bounded by the discrete-time cost no matter how the "discretization time parameter" is chosen.