Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem:  This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article. We can show this by using the closed graph theorem for set-valued functions, which says that for a compact Hausdorff range space "Y", a set-valued function "φ": "X"→2"Y" has a closed graph if and only if it is upper hemicontinuous and "φ"("x") is a closed set for all "x". Since all Euclidean spaces are Hausdorff (being metric spaces) and "φ" is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent. The Kakutani fixed point theorem can be used to prove the minimax theorem in the theory of zero-sum games. This application was specifically discussed by Kakutani's original paper. Mathematician John Nash used the Kakutani fixed point theorem to prove a major result in game theory. Stated informally, the theorem implies the existence of a Nash equilibrium in every finite game with mixed strategies for any number of players. This work later earned him a Nobel Prize in Economics. In this case:   In general equilibrium theory in economics, Kakutani's theorem has been used to prove the existence of set of prices which simultaneously equate supply with demand in all markets of an economy. The existence of such prices had been an open question in economics going back to at least Walras. The first proof of this result was constructed by Lionel McKenzie. In this case:   Kakutani's fixed-point theorem is used in proving the existence of cake allocations that are both envy-free and Pareto efficient. This result is known as Weller's theorem. The proof of Kakutani's theorem is simplest for set-valued functions defined over closed intervals of the real line. However, the proof of this case is instructive since its general strategy can be carried over to the higher-dimensional case as well. Let φ: [0,1]→2[0,1] be a set-valued function on the closed interval [0,1] which satisfies the conditions of Kakutani's fixed-point theorem. Let ("a""i", "b""i", "p""i", "q""i") for "i" = 0, 1, … be a sequence with the following properties:  Thus, the closed intervals ["a""i", "b""i"] form a sequence of subintervals of [0,1].