Geodesic
Yet Another Proof of the Minimax Theorem
Canadian mathematical bulletin, Tome 1 (1958) no. 2, pp. 97-100

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There are so many proofs of this theorem in the literature, that an excuse is necessary before exhibiting another. Such may be found by examining the proof given below for the following: it uses no matrices, almost no topology and makes little use of the geometry of convex sets; it applies equally well to the case where only one of the pure strategy spaces is finite; also there is no assumption that the payoff function is bounded. Thus it can provide a short route to the more general forms of the theorem. 
Peck, J.E.L. Yet Another Proof of the Minimax Theorem. Canadian mathematical bulletin, Tome 1 (1958) no. 2, pp. 97-100. doi: 10.4153/CMB-1958-011-x
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