Convergence of functions and their Moreau envelopes on Hadamard spaces

Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolík-Wijsman convergences of convex sets. As another application, we show that there exists a complete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that a sequence of functions converges in this metric if and only if it converges in the sense of Mosco.