In this paper we give a new proof of the existence of Puiseux series expansion of the limit discount equation of finite state : stochastic games. Unlike the original proof, due to Bewley and Kohlberg [Math. Oper. Res., 3 (1976), pp. 197-208], our proof is not algebraic and does not invoke Tarski’s principle. Instead we use only the theory of functions of complex variables and complex analytic varieties.