This paper addresses the problem of reachable set estimation for discrete-time switched systems under arbitrary switching. By introducing a novel conception called M-step sequence which is capable of characterizing all possible subsystem activation orders during M discrete-time steps, a Lyapunov function based approach is proposed to derive a set of bounding ellipsoids to estimate the reachable set. The proposed approach can cover the previous switched Lyapunov function approach and yields less conservativeness. Moreover, it can be shown that the M-step sequence method can also reduce the conservativeness in stability analysis for discrete-time switched systems under arbitrary switching in contrast to switched Lyapunov function method. Several numerical examples are provided to illustrate our approach.