For a function $g: X \times Y \rightarrow \mathbb R$ show that:
$$\sup_{y\in Y}\inf_{x\in X} g(x,y) \leq \inf_{x\in X}\sup_{y\in Y} g(x,y)$$
My attempt:
By definition: $\inf_{\bar x\in X} g(\bar x,y) \leq \sup_{\bar y\in Y} g(x,\bar y)$ for each $x,y$. In particular, this is true for the pair $(x,y)$ such that $y$ maximizes the LHS and $x$ minimizes the RHS. So the claim holds.
My difficulty
In my proof, I've assumed that the infimum and supremum are attained. This isn't necessarily the case if $X,Y$ are not closed or if the infimum/supremum is at $\infty$. What is a way to clean up this argument? My thought is to instead consider sequences that approach infimum and supremum, respectively. Is there some more direct way?
