  We show the strong convergence in arbitrary Sobolev norms of solutions of the
discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the
nonlinear Schr{\"o}dinger equation on the whole space. We restrict our
attention to the one and two-dimensional case, with a set of parameters which
implies global well-posedness for the continuous equation. Our proof relies on
the use of bilinear estimates for the Shannon interpolation as well as the
control of the growth of discrete Sobolev norms that we both prove.
