Let $X$ be a compact metric space, $\{\delta_n\}_{n=1}^{\infty}$ be a strictly monotonically decreasing sequence in $[0,1]$ converging to $0$, and $\{h_n\}_{n=1}^{\infty}$ be a uniformly convergence sequence of continuous functions on $X$ converging to $h:X\rightarrow [0,1]$. Distinguish a non-empty compact subset $A\subseteq X$. If:

- $h^{-1}[0]=A$,
- $h_n(A)\subseteq [0,\delta_n)$
*(for every $n$)*,

Can we guarantee that $$ I_{[0,\delta_n)}\circ h_n \mbox{ converges point-wise to } I_{\{0\}}\circ h? $$ If not, what additional conditions am I missing?

It seems that $I_{[0,\delta_n)}\circ h_n(A)=1$ for all $n$; which is good. So the convergence is uniform on $A$. However, I'm most worried about the pointwise convergence on $X-A$...this I can't manage to control...