In this paper the stochastic two person zero sum game of Shapley is considered, with metric state space and compact action spaces. It is proved that both players have stationary optimal strategies, under conditions which are weaker than those of Maitra and Parthasarathy (a.o. no compactness of the state space). This is done in the following way: we show the existence of optimal strategies first for the one-period game with general terminal reward, then for the n-period games (n=1,2,...); further we prove that the game over the infinite horizon has a value v, which is the limit of the n-period game values. Finally, the stationary optimal strategies are found as optimal startegies in the one-period game with terminal reward v.