Consider these sets
$$
A\equiv \cap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}
$$
$$
B_n\equiv \cap_{\delta>0}  \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}
$$
$$
C_n(L)\equiv \cap_{0<\delta\leq L}  \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\}=\{x \in X: d(p_n, [\ell(x), u(x)])\leq L\}
$$
$$
C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])\leq L_n\}
$$
where:

- $(p_n)_n$   is a sequence of  reals taking values in $[0,1]$.
- $\ell(\cdot)$ and $u(\cdot)$ are real  function taking values in $[0,1]$.
- $d\big(p_n,  [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
- $L_n$ is a sequence of strictly positive reals converging to zero as $n$ goes to infinity.


____________
Could you help me to show that
$$
 d_H(A, B_n)\rightarrow_{a.s.} 0,
$$
where 
$$
d_H(A, B_n)\equiv \max\{\sup_{x\in B_n}d(x,A), \sup_{x\in A}d(x, B_n)\},
$$
is the Hausdorff distance. Also, can we say something about $d_H(A, C_n(L))$ and $d_H(A, C_n(L_n))$?