Let $X$ be a smooth projective variety of dimension $\geq 2$. Fix an ample divisor $H$ and let $E$ be a $\mu$-semi-stable vector bundle of rank $2$ with $c_1(E)=0$. Assume $H^0(E)\neq 0$. I am looking for a proof of the following statement:
If $0\neq s\in H^0(E)$ vanishes no where, then $E$ is an extension of line bundles.
My attempt: Suppose such a section exists, then we have a surjection from $E^*\to \mathcal{O}_X$ (this follows from the definition of section of vector bundle), but I do not know how to proceed from here.
Thanks!