Let $(X,d_X)$ be a compact metric space and let $\{K_n\}_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x_n \in K_n$ we have $$ d_X(x_n,K)\leq \frac1{n}, $$ does this imply that $K$ is the Kuratowski lower limit ($\mathop{\mathrm{Li}}_{n \to \infty} K_{n}$) of the $K_n$; where the Kuratowksi limit limit is defined by: $$ \mathop{\mathrm{Li}}_{n \to \infty} K_{n} = \left\{ x \in X \left| \limsup_{n \to \infty} d(x, K_{n}) = 0 \right. \right\} $$

Let $(X,d_X)$ be a compact metric space and let $\{K_n\}_{n=1}^{\infty}$ be a collection of non-empty compact subsets. Let $K\subseteq X$ be compact. Then, if for every $x_n \in K_n$ we have $$ d_X(x_n,K)\leq \frac1{n}, $$ does this imply that $K$ is the Kuratowski lower limit ($\mathop{\mathrm{Li}}_{n \to \infty} K_{n}$) of the $K_n$, where the Kuratowksi limit limit is defined by $$ \mathop{\mathrm{Li}}_{n \to \infty} K_{n} = \left\{ x \in X \left| \limsup_{n \to \infty} d(x, K_{n}) = 0 \right. \right\} \;? $$