Prove that the set of continuation payoff vectors corresponding to all Nash equilibria is the…

Prove that the set of continuation payoff vectors corresponding to all Nash equilibria is the same in every proper subgame of a repeated game. The idea of the proof is to show more strongly that every proper subgame is strategically isomorphic, i.e., there is a one-to-one correspondence between the strategy spaces that preserves the payoffs. The simplest example is the map between the whole game and subgame: To map a strategy s for the whole game to its equivalent in the subgame