Consider the following sets: $$ \begin{aligned} & A_n\equiv \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)\leq \delta_n\Big\}\\ & A \equiv \Big\{ x\in X: \lim_{n\rightarrow \infty} d\big(p_n, [\ell(x), u(x)] \big)= 0\Big\} \end{aligned} $$ where:
- $X\subseteq \mathbb{R}$.
- $(p_n)_n$ is a sequence of numbers taking values in $[0,1]$.
- $\ell(\cdot)$ and $u(\cdot)$ are real function taking values in $[0,1]$.
- $(\delta_n)_n$ is a sequence of positive numbers going to $0$ as $n\rightarrow \infty$. In particular, $\delta_n\equiv \sqrt{\frac{1}{2n}\log(\frac{2}{\alpha})} $ with $\alpha\in (0,1)$.
- $d\left(a, E\right)\equiv \inf \left\{|a - y| : y \in E\right\}$.
Assume that $A$ is non-empty.
I would like to prove (or, disprove) that $d_H(A_n, A)\rightarrow 0$ as $n\rightarrow \infty$, where $d_H$ is the Hausdorff distance.
Comments: I know that the claim is wrong when $\delta_n\equiv 0$ for each $n$. See here for a counterexample.
More generally, I am trying to understand under which conditions the claim holds. The two sets seem so naturally "equivalent" as $n\rightarrow \infty$ that I am trying to exclude pathological cases where the claim does not hold.
As Amit said in his comment, it seems like you need to demand more, even if $A$ is non-empty.
For a simple example, take $\ell(x)=\max\{x-0.01,0\}$ and $u(x)=\min\{x+0.01,1\}$ with $X=[0,1]$.
Then $A_n= \{ x: \vert x-p_n\vert \leq \delta_n+0.01 \}$ if $0.1<\vert p_n \vert<0.9$.
If you take $p_n= \frac{1}{2}+(-1)^n \cdot 0.01$, then $A_{2n}=[\frac{1}{2}-\delta_n, \frac{1}{2}+0.02 +\delta_n]$ and $A_{2n+1}=[\frac{1}{2}-\delta_n-0.02, \frac{1}{2} +\delta_n]$.
If I am not mistaken, then $A=\{ \frac{1}{2} \}$, $A_{2n}\supseteq[\frac{1}{2}, \frac{1}{2}+0.02]$ and $A_{2n+1}\supseteq [\frac{1}{2}-0.02, \frac{1}{2}]$. Which means that $A_n$ is not a Cauchy sequence with respect to the Hausdorff metric.
So I think if $p_n$ does not converge and $\lim_{n\to \infty} \ell(p_n) \neq \lim_{n\to \infty} u(p_n)$, you can always have this case.
To avoid pathologies, you should probably assume that $u$ and $\ell$ are continuous and maybe also that $p_n\to p_\infty \in [0,1]$. Without assuming these there are probably many other convoluted counterexamples.