  We proved that any complete hypersurface in the Euclidean space
$\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to
be proper. As applications, we derive a counterpart of
Hoffman-Osserman-Schoen's result for $\lambda$-hypersurfaces, which gives an
affirmative answer to the Bernstein type problem proposed by Cheng-Wei
\cite{CW14b}.
