The nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as the discrete NLS (DNLS) on a periodic cubic lattice, which is a system of finitely many ODEs. We show that in two spatial dimensions, solutions to DNLS converge strongly in L two space to those to the corresponding continuum equation as the grid size goes to zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.