In mathematics, the max–min inequality is as follows: When equality holds one says that f, W, and Z satisfies a strong max–min property (or a saddle-point property).
f ( z , w ) = sin ( z + w )
illustrates that the equality does not hold for every function.
A theorem giving conditions on f, W, and Z which guarantee the saddle point property is called a minimax theorem.
Define
inf
by definition of the infimum being a lower bound.
by definition of the supremum being an upper bound.
making
an upper bound on
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,
inf
Putting all the pieces together, we get
inf
inf
inf
which proves the desired inequality.