  A well known result of H. Attouch states that the Mosco convergence of a
sequence of proper convex lower semicontinuous functions defined on a Hilbert
space is equivalent to the pointwise convergence of the associated Moreau
envelopes. In the present paper we generalize this result to Hadamard spaces.
More precisely, while it has already been known that the Mosco convergence of a
sequence of convex lower semicontinuous functions on a Hadamard space implies
the pointwise convergence of the corresponding Moreau envelopes, the converse
implication was an open question. We now fill this gap.
  Our result has several consequences. It implies, for instance, the
equivalence of the Mosco and Frol\'ik-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.
