  It is well known and readily seen that the maximum of $n$ independent and
uniformly on $[0,1]$ distributed random variables, suitably standardised,
converges in total variation distance, as $n$ increases, to the standard
negative exponential distribution. We extend this result to higher dimensions
by considering copulas. We show that the strong convergence result holds for
copulas that are in a differential neighbourhood of a multivariate generalized
Pareto copula. Sklar's theorem then implies convergence in variational distance
of the maximum of $n$ independent and identically distributed random vectors
with arbitrary common distribution function and (under conditions on the
marginals) of its appropriately normalised version. We illustrate how these
convergence results can be exploited to establish the almost-sure consistency
of some estimation procedures for max-stable models, using sample maxima.
