On convergence in the space of closed subsets of a metric space

We consider the space ${\rm clos}(X)$ of closed subsets of unbounded (not necessarily separable) metric space $(X, \varrho_{_X})$ endowed with the metric $\rho_{_X}^{\rm cl}$ introduced in [Zhukovskiy E.S., Panasenko E.A. // Fixed Point Theory and Applications. 2013:10]. It is shown that if any closed ball in the space $(X, \varrho_{_X})$ is totaly bounded,
then convergence in the space $\left({\rm clos}(X), \rho_{_X}^{\rm cl}\right)$ of a sequence $\{F_i\}_{i=1}^\infty$ to $F$ is equivalent to convergence in the sense of Wijsman, that is to convergence for each $x \in X$ of the distances $\varrho_{_X}(x, F_i)$ to $\varrho_{_X}(x, F).$