A difference equation with eventually periodic solutions

We show that every solution of the difference equation xn+1=max⁡{1xn,Axn−1},n=0,1,..., where the parameter A and the initial conditions x−1 and x0 are nonzero real numbers is eventually periodic with period 2, 3, or 4. We also give a precise description of the period in terms of A and the initial conditions.