  We consider the thresholding scheme, a time discretization for mean curvature
flow introduced by Merriman, Bence and Osher. We prove a convergence result in
the multi-phase case. The result establishes convergence towards a weak
formulation of mean curvature flow in the BV-framework of sets of finite
perimeter. The proof is based on the interpretation of the thresholding scheme
as a minimizing movement scheme by Esedoglu et. al.. This interpretation means
that the thresholding scheme preserves the structure of (multi-phase) mean
curvature flow as a gradient flow w. r. t. the total interfacial energy. More
precisely, the thresholding scheme is a minimizing movement scheme for an
energy functional that $\Gamma$-converges to the total interfacial energy. In
this sense, our proof is similar to the convergence results of Almgren, Taylor
and Wang and Luckhaus and Sturzenhecker, which establish convergence of a more
academic minimizing movement scheme. Like the one of Luckhaus and
Sturzenhecker, ours is a conditional convergence result, which means that we
have to assume that the time-integrated energy of the approximation converges
to the time-integrated energy of the limit. This is a natural assumption, which
however is not ensured by the compactness coming from the basic estimates.
