Game Theory Proof

I read an interesting theorem the other day which states that every two-person, finite, zero-sum game of perfect information is strictly determined. In other words, one of the players has a winning strategy: a strategy that guarantees a win no matter what the opponent does. Note that this does not apply to games that can end in a draw, such as chess. We will only be concerned with games that end with a win for one player and a loss for the opponent. The proof goes something like this:
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