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
$$
\mathop{\mathrm{Li}}_{n \to \infty} K_{n} = \left\{ x \in X \left| \limsup_{n \to \infty} d(x, K_{n}) = 0 \right. \right\} \;?
$$