Rational Learning Leads to Nash Equilibrium

Two players are about to play a discounted infinitely repeated bimatrix game. Each player knows his own payoff matrix and chooses a strategy which is a best response to some private beliefs over strategies chosen by his opponent. If both players' beliefs contain a grain of truth (each assigns some positive probability to the strategy chosen by the opponent), then they will eventually (a) accurately predict the future play of the game and (b) play a Nash equilibrium of the repeated game. An immediate corollary is that in playing a Harsanyi-Nash equilibrium of a discounted repeated game of incomplete information about opponents' payoffs, the players will eventually play an equilibrium of the real game as if they had complete information.