  Previously, many people have studied a stability of vector bundles of given
rank and Chern classes on algebraic varieties. Recently, we are interested in
the slope stability of the rank 2 Lazarsfeld-Mukai bundle $E_{C,Z}$ on a K3
surface $X$ associated to a very ample smooth curve $C$ on $X$ and a base point
free pencil $Z$ on $C$ with respect to $\mathcal{O}_X(C)$. In this paper, we
will give a sufficient condition for such a Lazarsfeld-Mukai bundle $E_{C,Z}$
to be $\mathcal{O}_X(C)$-slope semistable by ACM line bundles with respect to
$\mathcal{O}_X(C)$.
