Let X and Y be nonempty sets and let h : X Y R have bounded range in R. Let f : X

Let X and Y be nonempty sets and let h : X × Y †’ R have bounded range in R. Let f : X †’ R and g : Y †’ R be defined by
F(x) := sup{h(x; y) : y ˆˆ Y}; g(y) := inf{h(x; y) : x ˆˆ X}:
Prove that
Sup{g(y) : y ˆˆ Y} We sometimes express this by writing

Exercises 9 and 10 show that the inequality may be either an equality or a strict inequality.

sup(x,y) ^ inf suph(x,y).