Minimax theorem as a consequence of Hahn Banach theorem

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Please I have difficulty relating the minimax theorem to the Hahn Banach theorem in functional Analysis...minimax is a consequence of Hahn Banach but I just can't see it... Please I need someone who's good in analysis and optimization/game theory to help me.

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For a proof of the standard Sion version, see:

p. 5-6, Mertens, Jean-François, Sylvain Sorin, and Shmuel Zamir. Repeated Games. Vol. 55. Cambridge University Press, 2015.