Let σ − i be a mixed strategy profile of all players except for player i. We can define a best response correspondence for player i, bi. bi is relation from the set of all probability distributions over opponent player profiles to a set of player i's strategies, such that each element of bi(σ − i) is a best response to σ − i. Define
http://en.wikipedia.org/math/98875e4...e75704079a.png One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a σ * such that
http://en.wikipedia.org/math/705cd11...60404db399.png Since b(σ * ) represents the best response for all players to σ * , the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium.