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Distance between two sets different from zero

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ A_n\equiv \{x \in X: d(p_n, [\ell(x), u(x)])=0\} $$ 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. Can you give me a simple counterexample of why $d_H(A,A_n)\neq 0$?

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