We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in<img src="/fulltext-image.asp?format=htmlnonpaginated&src=DK5711T02N119611_html\13_2005_Article_1601_TeX2GIFIEq1.gif" border="0" alt="$$ \mathbb{R}^{m}(m=4, 5 ) $$" /> with finite index must be minimal.