  Let R be a commutative noetherian local ring, and M a finitely generated
R-module of infinite projective dimension. It is well-known that the depths of
the syzygy modules of M eventually stabilize to the depth of R. In this paper,
we investigate the conditions under which a similar statement can be made
regarding dimension. In particular, we show that if R is equidimensional and
the Betti numbers of M are eventually non-decreasing, then the dimension of any
sufficiently high syzygy module of M coincides with the dimension of R.
