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.
- $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
- 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:
Show that $$ (1) \Pr(\lim_{n\rightarrow \infty} d(x, A)=0)=1\quad \forall x\in A_n $$
Show that $$ (2) \Pr(\lim_{n\rightarrow \infty} d(x, A_n)=0)=1\quad \forall x\in A $$
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?