Assume you have a real valued function $f(x,y)$ defined on some domain $X\times Y$.
When people write $\sup\limits_{x \in X}\sup\limits_{y\in Y} |f(x,y)|$ is that actually the same as
$\sup\limits_{(x,y) \in X \times Y } |f(x,y)|$?
Such that
$\sup\limits_{y\in Y}\sup\limits_{x \in X} |f(x,y)|=\sup\limits_{x \in X}\sup\limits_{y\in Y} |f(x,y)|$
For instance in the Arzela Ascoli theorem one requires among a other condition for a set $\lbrace f_1,f_2,\ldots\rbrace $ to be relatively compact in $C(X, \mathbb{R})$ ($X$ for example compact metric space) that $\sup\limits_{n \in \mathbb{N}} \sup\limits_{x\in X} |f_n(x)|<\infty$.
Is that the same as $\sup\limits_{x\in X} \sup\limits_{n \in \mathbb{N}} |f_n(x)|<\infty$