Will the outcome of a game always be a Nash Equilibrium?

Consider this game between two players. This game has two Nash Equilibria: (U, C) and (D, R). Suppose we ask the players to play this game once. What should our prediction of the game’s result be?

If Player 2 thinks Player 1 will play ‘D’ (since (D, R) gives player 1 gives higher payoff), then he’d play R. However, if it is resonable to think that Player 1 will play D since (D, R) gives him higher pay off, it is also reasonable to think that Player 1 will want to play C since the NE (U, C) gives him a higher payoff.

With that said, what is stopping the players from predicting the other player’s move wrong? Suppose: Player 1 plays D hoping to get to the Nash Equilibrium (D, R). But Player 2 plays C hoping to the Nash Equilibrium (U, C). When they do this, they end up at (D, C)!

If the outcome of a single stage game will not always be the Nash Equilibrium, then why do we take it for a given that in the final stage of a repeated game, the players will always pick the Nash Equilibrium?