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



Criterion for Kuratowski Limit Inferior
The answer is no, or what am I missing. Let $X = K = \{0,1\}$ and $K\_n = \{0\}$, $n \in \mathbb{N}$ with $d\_X(0,1) = 1$. Then $\text{LI}\_{n \to \infty} K\_n = \{0\} \not= K$.


