https://www.wikiwand.com/en/Max%E2%80%93min_inequality

$\sup _ { z \in Z } \inf _ { w \in W } f ( z , w ) \leq \inf _ { w \in W } \sup _ { z \in Z } f ( z , w )$

(note that operators are written in the opposite order in which they act..) What inequality says is: If sup acts first and then inf, we get smaller or equal value than if inf acts first and then sup basically whoever acts last sup or inf wins (whoever laughs last laughs the best) this is because after sup act, and then inf acts, what we get is a lower bound on how good sup could act no matter what inf did. No matter what inf does, doing what sup does in the first case gives a value at least as big as sup inf, so that what sup can do after inf (inf sup) is at least that

Proof idea: You can show this from the fact that taking max or min of both sides of an inequality, preserves the inequality! (can be shown in one line). Then we can take the min_w f(z,w) <= f(z,w') for any w'. Then take the max_z of both sides. And then because this holds for any w', one can take the min of the RHS and it still holds!