  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 generalized nonlinearities,
the latter is exemplified by generic power-law and saturable ones. For
illustration of the applicability of the approach a brief discussion of the
results of numerical studies is included, showing that when 1-soliton solution
of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear
Schr\"odinger system with cubic and saturable nonlinearity, its long-term
persistence. Thereby excellent agreement of the numerical findings with the
theoretical predictions are obtained.
