How is it possible to prove the following inequality?
$\inf_{x \in [a,b]} \{\liminf_{n \to \infty} f_n(x)\} \le \liminf_{n \to \infty} \{\inf_{x \in [a,b]} f_n(x)\}$
I tried to solve this problem by substituting $"\liminf"$ for $"\sup"$ and $"\inf"$ but I did not be able to solve it.
This is false. All we need for a counterexample is a sequence of functions $f_n$ so that for each $x$, the sequence $f_n(x)$ is eventually $1$, but for each $n$, $f_n$ has a zero in the interval $[a, b]$. In this way $\inf_x f_n(x)=0$ and $\liminf_n f_n(x)=1$ and your inequality would imply $1\leq 0$.
We can just take, say, $f_n(x)$ equal to $1$ everywhere, except at $x=a+\frac {b-a}n$ where it's zero.