Let $X$ be a compact metric space and $\{h_n\}_{n=1}^{\infty}$ be a uniformly convergence sequence of continuous functions on $X$ converging to $h:X\rightarrow \mathbb{R}$. Let $A_n=[0,\delta_n]$ be a nested sequence of closed subsets of $\mathbb{R}$ for which $[0]=\cap_{n=1}^{\infty} A_n$. Suppose also that $A\subseteq h_n^{-1}[A_n]$ for each $n$.
Then, can we guarantee that $$ I_{A_n}\circ h_n \mbox{ converges point-wise to } I_{A}\circ h? $$
** What I have:** $I_{A_n}$ converges point-wise $I_A$, pretty much by definition. But where do we go from here?
False. Let $X=[0,1]$ , $h_n(x)=3\delta_n x$ and $h(x)=0$ for all $x$. Then $I_A(h(\frac 1 2))=1$ but $I_A(h_n(\frac 1 2))=0$ for all $n$.