 Solution

Take the compact Riemann surface $X={\mathbb{P}}^1$ and the holomorphic vector bundle $E={\mathcal{O}}_X\oplus {\mathcal{O}}_X$. For every positive integer $m$, use homogeneous coordinates $[Z_0:Z_1]$ and the mapThe two sections $Z_0^m,Z_1^m$ never vanish together on ${\mathbb{P}}^1$. Thus this map has rank one at every point; locally one of its two components is invertible in suitable line-bundle frames, so its image is a genuine holomorphic subbundle and its quotient is a locally free sheaf. More explicitly, there is an exact sequenceExactness follows fibrewise from the absence of a common zero and hence also locally as a sequence of vector bundles. The image $F_m$ is isomorphic to ${\mathcal{O}}_X(-m)$ and has rank one, soThus there need not be a lower bound for subbundle slopes, even for a fixed trivial rank-two bundle. Merely multiplying a section of one fixed subline by a function with zeros would not suffice: that could give only a subsheaf with a torsion quotient. The nowhere-simultaneously-zero pair above avoids that problem and realizes unbounded negative degrees of line subbundles.