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

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



Let
$$
d_H(A, B)\equiv \max\{\sup_{x\in B}d(x,A), \sup_{x\in A}d(x, B)\},
$$
denote the Hausdorff distance. Is $d_H(A,C_n(L_n))= 0$?