Max–min inequality
In mathematics, the max–min inequality is as follows:
When equality holds one says that,, and satisfies a strong max–min property. The example function illustrates that the equality does not hold for every function.
A theorem giving conditions on,, and which guarantee the saddle point property is called a minimax theorem.
Proof
Define For all, we get for all by definition of the infimum being a lower bound. Next, for all, for all by definition of the supremum being an upper bound. Thus, for all and, making an upper bound on for any choice of. Because the supremum is the least upper bound, holds for all. From this inequality, we also see that is a lower bound on. By the greatest lower bound property of infimum,. Putting all the pieces together, we getwhich proves the desired inequality.