Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We endow $\mathcal C$ with the Hausdorff metric $d_H$. It's well-known that the topology of $\mathcal C$ depends on that of $E$, not on $d$. One example to illustrate this independece is as follows. Let $A,A_n \in \mathcal C$ such that $A_n \to A$ in $d_H$. Then $$ \lim_n A_n = \{ \lim x_n \mid (x_n) \in (A_n)\text{ convergent}\}. $$
This means the limit point $A$ of the sequence $(A_n)$ is characterized only by the convergent sequences in $E$. My question is the following.
Is there a simple characterization of $d_H(A_n, A) \to 0$ in terms of convergent sequences in $E$?
