Does $\lim_{n \to \infty}\inf_{x\in X} f_n (x) = \inf_{x\in X} f (x)$ hold where $f_n$ is bounded Lipschitz continuous such that $f_n \uparrow f$?

Question

Let $X$ be a Polish space, $f_n: X \to \mathbb R$ bounded Lipschitz continuous, and $f: X \to \mathbb R \cup \{+\infty\}$ such that $f_n \uparrow f$ pointwise. Then $$ A :=\lim_{n \to \infty}\inf_{x\in X} f_n (x) \le B:= \inf_{x\in X} f (x). $$

Does the reverse inequality $A \ge B$ hold? Thank you so much for your elaboration.

Answer 1

A counterexample is the sequence of functions $f_n:\mathbb{R}\to\mathbb{R}$ given by $$f_n(x)=\frac{1}{\pi}\arctan(n-x)+\frac{1}{2},$$ which increases pointwise to $f:\mathbb{R}\to\mathbb{R}$, $f(x)=1$ for all $x\in\mathbb{R}$. However, we have $$\inf_{x\in\mathbb{R}}f_n(x)=0$$ for all $n\in\mathbb{N}$, but $$\inf_{x\in\mathbb{R}}f(x)=1.$$