Lets say we have a function $f:X \times Y \to \mathbb R$. Is it always true that
$$ \liminf_{y \to b} \limsup_{x \to a} f(x,y) \geq \limsup_{x \to a} \liminf_{y \to b} f(x,y)$$ ?
This question was inspired by the fact that we always have $$\inf_{y \in Y} \sup_{x \in X} f(x,y) \geq \sup_{x \in X} \inf_{y \in Y}f(x,y)$$
I have been trying to use the latter to prove the former, but have had no success, so I am starting to doubt whether or not it is true. Can someone shed some insight?
Turns out this is false. Consider $f(x,y)=[x>y]$ and $a=b=\infty$.