最大最小不等式
g ( z ) ≜ inf w ∈ W f ( z , w ) {\displaystyle g(z)\triangleq \inf _{w\in W}f(z,w)} と定義する。
⟹ g ( z ) ≤ f ( z , w ) , ∀ z , w {\displaystyle \implies g(z)\leq f(z,w),\forall z,w}
⟹ sup z g ( z ) ≤ sup z f ( z , w ) , ∀ w {\displaystyle \implies \sup _{z}g(z)\leq \sup _{z}f(z,w),\forall w}
⟹ sup z inf w f ( z , w ) ≤ sup z f ( z , w ) , ∀ w {\displaystyle \implies \sup _{z}\inf _{w}f(z,w)\leq \sup _{z}f(z,w),\forall w}
⟹ sup z inf w f ( z , w ) ≤ inf w sup z f ( z , w ) ◻ {\displaystyle \implies \sup _{z}\inf _{w}f(z,w)\leq \inf _{w}\sup _{z}f(z,w)\qquad \square }