The author considers a difference equation of the form \[ x_{n+1} - x_n= g(n,x_{n-k_0},\dots,x_{n-k_m}),\quad n=0,1,\dots \tag{1} \] where \(k_0,k_1, \dots, k_m\) are nonnegative integers, \(g(n,u_0, \dots, u_m)\) is continuous and nonincreasing in each of its arguments \(u_0, \dots, u_m\), and \(g(n,0, \dots, 0) \equiv 0\) for all \(n\). Sufficient conditions for every solution of (1) to tend to zero are given. These results are applied to obtain global attractivity of delay differential equations with piecewise constant arguments.