Good night, I'm reading Sion's Minmax Theorem:
and its proof:
Could you please confirm if my understanding why $\max_{x \in X} \inf _{y \in Y^{\prime}} g(x, y) < v < \inf _{y \in Y^{\prime} \sup_{x \in X}} g(x, y)$ is correct?
My attempt:
- $\max_{x \in X} \inf _{y \in Y^{\prime}} g(x, y) < v$
It suffices to show that for all $x \in X$, $\inf_{y \in Y^{\prime}} g(x, y)<v$. To prove $\inf_{y \in Y^{\prime}} g(x, y)<v$, it suffices to show that $\exists y \in Y': g(x,y) < v$.
Because $(X_y)_{y \in Y_0}$ is a finite subcover of $X$ and $Y_0 \subseteq Y'$, there is $\bar y \in Y'$ such that $x \in X_{\bar y}$. Then $g(x,\bar y) < v$. This completes the proof.
- $v < \inf_{y \in Y^{\prime}} \sup_{x \in X} g(x, y)$
It follows from $Y' \subseteq Y$ that $\inf _{y \in Y} \sup _{x \in X} g(x, y) \le \inf _{y \in Y'} \sup _{x \in X} g(x, y)$. On the other hand, $v < \inf _{y \in Y} \sup _{x \in X} g(x, y)$. Hence $v < \inf_{y \in Y^{\prime}} \sup_{x \in X} g(x, y)$.