  We prove that the solutions to the discrete Nonlinear Schr\"odinger Equation
(DNLSE) with non-local algebraically-decaying coupling converge strongly in
$L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear
Schr\"odinger Equation (FNLSE), as the discretization parameter tends to zero.
The proof relies on sharp dispersive estimates that yield the Strichartz
estimates that are uniform in the discretization parameter. An explicit
computation of the leading term of the oscillatory integral asymptotics is used
to show that the best constants of a family of dispersive estimates blow up as
the non-locality parameter $\alpha \in (1,2)$ approaches the boundaries.
