  We consider the K\"ahler-Ricci flow on a Fano manifold. We show that if the
curvature remains uniformly bounded along the flow, the Mabuchi energy is
bounded below, and the manifold is K-polystable, then the manifold admits a
K\"ahler-Einstein metric. The main ingredient is a result that says that a
sufficiently small perturbation of a cscK manifold admits a cscK metric if it
is K-polystable.
