  While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear
Schr\"odinger equation, which is more significant for physical applications, is
not. We prove closeness of the solutions of both systems in the sense of a
"continuous dependence" on their initial data in the $l^2$ and $l^{\infty}$
metrics. The most striking relevance of the analytical results is that small
amplitude solutions of the Ablowitz-Ladik system persist in the Discrete
Nonlinear Schr\"odinger one. It is shown that the closeness results are also
valid in higher dimensional lattices as well as for generalised nonlinearities.
For illustration of the applicability of the approach, a brief numerical study
is included, showing that when the 1-soliton solution of the Ablowitz-Ladik
system is initiated in the Discrete Nonlinear Schr\"odinger system with cubic
and saturable nonlinearity, it persists for long-times. Thereby excellent
agreement of the numerical findings with the theoretical predicti ions is
obtained.
