limit infimum and limit of a sequence of functions
Solution 1:
If you mean limit inferior instead of limit infimum, they are equal. For every $x$, $(f_n(x))_n$ is a sequence. When a sequence has a limit, than limit is equal to the liminf (and equal to the limsup), so $\text{liminf}_{n \to \infty} f_n(x) = \lim_{n\to\infty}f_n(x) = f(x).$
If you mean the limit for $n\to \infty$ of the infimum of the set $\{f_k(x)\}_k,$ then, as someone else already pointed out, that would just be the limit of a constant sequence. This limit would equal $\inf_n f_n(x),$ but I don't believe that this makes any sense.