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 $d_H(A,A_n)\neq 0$?

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 $d_H(A,C_n(L_n))= 0$?

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])\leq L_n\} $$ 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 $[0,1]$.
• $d\big(p_n, [\ell(x), u(x) ] \big):= \inf \big\{|p_n - y| : y \in [\ell(x), u(x) ] \big\}$.
• $L_n$ is a sequence of strictly positive reals converging to zero as $n$ goes to infinity.

Does $d_H(A, C_n(L_n))$ go to zero, where $$ d_H(A, B)\equiv \max\{\sup_{x\in B}d(x,A), \sup_{x\in A}d(x, B)\}, $$ is the Hausdorff distance?

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 $d_H(A,A_n)\neq 0$?