Summary: Let \((X,d)\) be a metric space and \(\text{CL}(X)\) the family of all nonempty closed subsets of \(X\). We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric \(d\) forces \(X\) to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover, we prove that \((X,d)\) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on \(\text{CL}(X)\) coincide.