Is this a correct proof of the Max-Min Inequality? And if not how could it be improved? Thank you for your help.
Given $g\colon X\times Y\to \mathbb{R}$, we note that for any $\tilde{x} \in X$, $\tilde{y}\in Y$ that
$$\inf_{x\in X}g(x,\tilde{y}) \leq g(\tilde{x},\tilde{y}) \leq \sup_{y\in Y}g(\tilde{x},y)$$ so for arbitrary $x\in X$, $y\in Y$ $$\inf_{x\in X}g(x,y) \leq \sup_{y\in Y}g(x,y)$$
Taking the supremum over all $y\in Y$ leaves the right-hand side unaffected so $$\sup_{y\in Y}\left(\inf_{x\in X}g(x,y)\right)\leq \sup_{y\in Y}g(x,y)$$ Finally take the infimum over all $x\in X$ yields the desired result $$\sup_{y\in Y}\left(\inf_{x\in X}g(x,y)\right)\leq \inf_{x\in X}\left(\sup_{y\in Y}g(x,y)\right)$$