In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions in [17]. Its solution (v;m) can be obtained as the limit of the solutions of the second order mean field game problems, when the noise parameter tends to zero (see [17]). We propose a semi-discrete infitime approximation of the system and, under natural assumptions, we prove that it is well posed and that it converges to (v;m) when the discretization parameter tends to zero.