  We show that, given an immortal solution to the Ricci flow on a closed
manifold with uniformly bounded curvature and diameter, the Ricci tensor goes
to zero as t goes to infinity. We also show that if there exists an immortal
solution on a closed 3-dimensional manifold such that the product of the square
of the diameter with the norm of the curvature tensor is uniformly bounded,
then the solution must be of type III.
