  In \cite{D3}, Donaldson defines a dynamical system on the space of
Fubini-Study metrics on a polarized compact K\"ahler manifold. Sano proved that
if there exists a balanced metric for the polarization, then this dynamical
system always converges to the balanced metric (\cite{S}). In \cite{DKLR},
Douglas, et. al., conjecture that the same holds in the case of vector bundles.
In this paper, we give an affirmative answer to their conjecture.
