Max–min inequality
In mathematics, the max–min inequality is as follows:

For any function 
 

When equality holds one says that , , and  satisfies a strong max–min property (or a saddle-point 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 get

 

which proves the desired inequality. References

See also
 Minimax theorem

Mathematical optimization
Inequalities