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: 

- $X\subseteq \mathbb{R}$.
- $(p_n)_n$   is some sequence of reals 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.
____________
**My attempt:**
 To show the desired claim, I would proceed as follows:

1. Show that
$$
(1) \Pr(\lim_{n\rightarrow \infty} d(x, A)=0)=1\quad \forall x\in A_n
$$

2. Show that
$$
(2) \Pr(\lim_{n\rightarrow \infty} d(x, A_n)=0)=1\quad \forall x\in A
$$

3. Show that (1)+(2) implies that $d_H(A, A_n)$ goes to zero.


I am having trouble formally showing (1) and (2). Intuitively, they must hold. Could you advise?