  In this paper, we study four dimensional hypersurface M^4_r with proper mean
curvature vector field (i.e. \Delta\vec{H} is proportional to \vec{H}) in
pseudo-Riemannian space form N^5_s(c), and show that it has constant mean
curvature, and give the range of this constant. As an application, we get that
biharmonic hypersurfaces in N^5_s(c) are minimal in some specific cases, which
partially confirms B.-Y. Chen's conjecture.
