Construction of a quotient line bundle of a semistable vector bundle

Question

I want to prove the following Lemma, but I'm stuck.

Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $g\geq3$ and $E$ be a semistable vector bundle of rank $n$ and degree $d$ with $0\leq d\leq 2n(g-1)$ over $X$
If $h^1(E)>0$ and $h^0(E)>0$, we can construct a quotient line bundle $E\rightarrow L\rightarrow 0$ with $\operatorname{deg}(L) \leq2g-2$.

I think $h^1(E)>0$ is the key condition and this is a dumb question.
Thanks in advance.

Answer 1

$H^1(E)\neq 0$ implies by duality, $H^0(E^*\otimes K)\neq 0$. Take a section of $E^*\otimes K$ and saturate it to get a line subbundle $M\subset E^*\otimes K$ and $H^0(M)\neq 0$ and thus $\deg M\geq 0$. Dualize to get a surjection $E\to M^{-1}\otimes K=L$ and notice that $\deg L\leq 2g-2$ since $\deg M^{-1}\leq 0$.