Criterion for Kuratowski Limit Inferior

Question

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

Answer 1

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 $\text{LI}_{n \to \infty} K_n = \{0\} \not= K$.