We consider the asymptotic behavior of bounded solutions of the difference equations of the form x(n+1)=Bx(n) + y(n) in a Banach space X, where n=1,2,..., B is a linear continuous operator in ${mathbb X}$, and (y(n)) is a sequence in ${mathbb X}$ converging to 0 as n . An obtained result with an elementary proof says that if then every bounded solution x(n) has the property that . This result extends a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof based on Complex Functions are furthered to study the stability of solutions of the discrete system. A discussion on further extensions is also given.