Consider the following sets:
$$
A =  \Big\{ x\in X: \Pr\bigg(\lim_{n \to \infty}d\big(p_n,  [\ell(x), u(x) ] \big)= 0\bigg)=1 \Big\},
$$
and
$$
A_n =  \Big\{ x\in X:  d\big(p_n,  [\ell(x), u(x) ] \big)= 0\Big\},
$$
where: 

- $\Pr$ denotes probability.
- $X\subseteq \mathbb{R}$.
- $(p_n)_n$   is a sequence of  random variables taking values in $[0,1]$.
- $\ell(\cdot)$ and $u(\cdot)$ are real  function taking values in $[0,1]$.
- $d\big(p_n,  [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
- A is non-empty.


Could you help me to show that
$$
 d_H(A, A_n)\rightarrow_{a.s.} 0,
$$
where 
$$
d_H(A, A_n)\equiv \max\{\sup_{x\in A_n}d(x,A), \sup_{x\in A}d(x, A_n)\},
$$
is the Hausdorff distance.
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**Additional details which may be helpful:**

- Note that $A$ is assumed non-empty.

- Also note that $(p_n)_n$ does not necessarily converge.

- Lastly, I have shown that, $\forall x\in A$ and $\delta>0$, $\Pr(x\in A_n(\delta))\geq 1-\frac{2}{\exp(2n \delta^2)}$,
where
$$
A_n(\delta) =  \Big\{ x\in X:  d\big(p_n,  [\ell(x)-\delta, u(x)+\delta ] \big)= 0\Big\}.
$$
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