While I am reading Tirole Game Theory Book , I read the following statement
If a single strategy profile survives iterated deletion of strictly dominated strategies, then it is a Nash equilibrium of the game.
How can I prove this statement?
I fact, I know that the converse is not true. That is, any Nash-equilibrium profile must put weight only on strategies that are not strictly dominated (or more generally, do not survive iterated deletion of strictly dominated strategies), because a player could increase his payoff by replacing a dominated strategy with one that dominates it. [However, Nash equilibria may assign positive probability to weakly dominated strategies]
But I cannot proceed anything for the other side of the statement.
I will be appreciated for all helps. Thank you.