This is the most interesting real analysis question I have run into thus far (that I understand):
For all $n\in\mathbb{N}$, let $f_n(x)$ and $f(x)$ be one-to-one continuous functions such that $B=\bigcap_{n=1}^{\infty}f_n(A)$ is a nonempty interval. If $f_n(x)$ converges pointwise to $f(x)$ on $A$, then $f_n^{-1}(x)$ converges pointwise to $f^{-1}(x)$ on $B$.
Do you think this is a valid assertion, or are other conditions necessary for it to hold water?
Also, how could one go about proving it? Thanks in advance!
Edit 1: Because $f_n\to f$, is it safe to assume that $f:A\to B$? If so, then could I conclude that $f$ is bijective on $A$ and therefore $f^{-1}:B\to A$? I still feel that this is not rigorous enough, however.