We prove an analogue of a classical asymptotic stability result of standing waves of the Schrodinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi and it involves a discrete Schr\"odinger operator H=-\Delta +q . The decay rates on the potential are less stringent than in Mizumachi. We also prove |e^{itH}(n,m)|\le C \langle t \rangle ^{-1/3} for a fixed $C$
