A concept of dynamic stability in infinitely repeated games with discounting is presented. For this purpose, one modification of the available theory is needed: we need to relax the assumption that the game starts in a given period. Under this new framework, we propose stable strategies such that a folk theorem with an additional stability requirement still holds. Under these strategies, convergence to the long run outcome is achieved in a finite number of periods, no matter what actions or deviations have been played in the past. Hence, we suggest a way in which a player can build up his reputation after a deviation.