Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It is true, that if $g$ is continuous and $f_n\to f$ compactly, i.e. uniformly on every compact set, then also the inverse functions $g_n$ converge to $g$ compactly.
In Uniform convergence of functions, Spring 2002 they gave a nice proof for a similar result, using uniform convergence of $f_n$ and uniform continuity of $f$ to conclude that $g_n$ converge uniformly. The idea was to show that $f\circ g_n$ converges to $f\circ g$ uniformly and then conclude that also $g_n = g\circ f\circ g_n$ converge uniformly to $g = g\circ f\circ g$.
I would be interested if it is possible to adapt this prove to compact convergence. The main problem I see is the following: Showing $f\circ g_n$ converges to $f\circ g$ compactly is equivalent to showing $f\circ g_n$ converges to $f_n\circ g_n$ compactly. But for fixed compact set $K\subset Y$ I do not see how we can guarantee that all $g_n(K)$ stay in the same compact set $L\subset X$. Hence, we can not apply compact convergence of $f_n$. Are there any ideas for finding the compact set $L$?