  For a holomorphic vector bundle $E$ over a polarised K\"ahler manifold, we
establish a direct link between the slope stability of $E$ and the asymptotic
behaviour of Donaldson's functional, by defining the Quot-scheme limit of
Fubini-Study metrics. In particular, we provide an explicit estimate which
proves that Donaldson's functional is coercive on the set of Fubini-Study
metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein
metrics implying slope stability.
