  We present a new approach to the theory of asymptotic properties of solutions
of difference equations. Usually, two sequences $x,y$ are called asymptotically
equivalent if the sequence $x-y$ is convergent to zero i.e., $x-y\in c_0$,
where $c_0$ denotes the space of all convergent to zero sequences. We replace
the space $c_0$ by various subspaces of $c_0$. Our approach is based on using
the iterated remainder operator. Moreover, we use the regional topology on the
space of all real sequences and the `regional' version of the Schauder fixed
point theorem.
