  The study of nonlinear Schr\"odinger-type equations with nonzero boundary
conditions define challenging problems both for the continuous (partial
differential equation) or the discrete (lattice) counterparts. They are
associated with fascinating dynamics emerging by the ubiquitous phenomenon of
modulation instability. In this work, we consider the discrete nonlinear
Schr\"odinger equation with linear gain and nonlinear loss. For the infinite
lattice supplemented with nonzero boundary conditions which describe solutions
decaying on the top of a finite background, we give a rigorous proof that for
the corresponding initial-boundary value problem, solutions exist for any
initial condition, if and only if, the amplitude of the background has a
precise value $A_*$ defined by the gain-loss parameters. We argue that this
essential property of this infinite lattice can't be captured by finite lattice
approximations of the problem. Commonly, such approximations are defined by
lattices with periodic boundary conditions or as it is shown herein, by a
modified problem closed with Dirichlet boundary conditions. For the finite
dimensional dynamical system defined by the periodic lattice, the dynamics for
all initial conditions are captured by a global attractor. Analytical arguments
corroborated by numerical simulations show that the global attractor is
trivial, defined by a plane wave of amplitude $A_*$. Thus, any instability
effects or localized phenomena simulated by the finite system can be only
transient prior the convergence to this trivial attractor. Aiming to simulate
the dynamics of the infinite lattice as accurately as possible, we study the
dynamics of localized initial conditions on the constant background and
investigate the potential impact of the global asymptotic stability of the
background with amplitude $A_*$ in the long-time evolution of the system.
