For switched discrete-time systems, switching behavior always affect the finite-time stability property, which was neglected by most previous research. This paper investigated the problem of finite-time boundness of $H_{infty}$ filtering for switched discrete-time systems. Sufficient conditions which can ensure finite-time bounded and $H_{infty}$ filtering finite-time boundness under arbitrary switching are derived. Based on the results of finite-time boundness and stochastic character, the closed-loop system trajectory stays within a prescribed bound. An example is given to illustrate the efficiency of the proposed method.