Let $C$ be a smooth projective curve and $E$ a vector bundle of rank $r$ on $C$. We say that $E$ is nef/ample if $\mathcal{O}_{\mathbb{P}(E)}(1)$ is so. Equivalently (see Hartshorne's papers on 'Ample vector bundles' and 'Ample vector bundles on curves'), $E$ is ample if and only if for any coherent $F$, $S^m(E)\otimes F$ is globally generated for all $m\geq n_0$.

The statement I'm slightly stuck on is the following comment in a paper of Fujita 'On Kahler fibre spaces over curves': If $C$ has genus $g\geq2$ and $H^1(C, E)=0$ for some vector bundle $E$, then $E$ is ample.

This follows easily in the rank $1$ case from Riemann-Roch. I suspect the general case will also be easy but I have been through Lazarsfeld's book and the standard references with no luck so far. Any help appreciated!