$\sup(\lim) \le \lim(\sup)$?

Question

$\sup_{x,y\in A} \lim_{n\to\infty} f_n(x,y) \le \lim_{n\to\infty} \sup_{x,y\in A} f_n(x,y)$ ?

Proof) $f_n\le \sup f_n.\\ \lim f_n\le \lim(\sup f_n)\\ \sup(\lim f_n)\le \lim(\sup f_n)$

I think that if I want this proof to be correct, I need some conditions. (such as $\lim f_n, \lim(\sup f_n)$ exists...)

Does $\lim_{n \to \infty} \sup_{x \in X} f_n(x) = \sup_{x \in X} \lim_{n \to \infty} f_n(x)$?

I read this question and all the counterexamples in the answers were not enough to dispute sup lim $\le$ lim sup (Although they dispute sup lim = lim sup)

In summary, what I like to know is,

1.$\sup(\lim) \le \lim(\sup)$?(with what condition?)

2.What are the required conditions for it to be correct?

Thank you for reading.