It is clear that stable vector bundles over a projective scheme are simple, i.e. $\operatorname{Hom}(F,F) \cong \mathbb{C}$.
Let $X$ now be an elliptic curve.
Then every simple vector bundle is stable. How does one prove stability?
I assume one would approach this by contradiction. So let $E\subsetneq F$ be a subbundle of $F$ such that $\mu(E) > \mu (F)$. As we are over an elliptic curve, we know that $\operatorname{Hom}(E,F) = 0$. But what now?
The embedding $E \subsetneq F$ gives a morphism $\varphi \in \operatorname{Hom}(E,F) = 0$, so $\varphi = 0$. Since $\varphi$ is injective, $E = 0$. This shows that $F$ is stable, because you have to check the slope condition on all non-zero subsheaves.