# Stability of holomorphic vector bundles

I'm reading the book "R. O. Wells Jr. - Differential Analysis on Complex Manifolds" and on page 247 the author claims two things about the stability of holomorphic vector bundles that I'm struggling to prove:

Here I'm talking about holomorphic vector bundles over a Riemann surface.

1)Every polystable bundle is in particular semistable.

2)If $$F\subset E$$ is a subbundle s.t $$\mu(F)=\mu(E)$$, then $$E/F$$ is semistable with $$\mu(E)=\mu(E/F)$$.

I think it should be easy to see, but as I'm not very familiar with this subject, I don't know where to start. Does anyone have any references or suggestions?

For Question 1, let's take $$E=E_1\oplus E_2$$, and let $$\mu$$ be the slope of $$E$$ (which agrees with the slope of $$E_1$$ and $$E_2$$). Let $$F\subset E$$ be a subbundle and let $$p:=\mathrm{pr_1}|_F:F\to E_1$$ be the projection on the first factor. There is an exact sequence $$0\to \ker(p)\to F \to \mathrm{Im}(p) \to 0$$ Note that $$\ker(p) \subset E_2$$ and $$\mathrm{Im}(p) \subset E_1$$. By semistability of $$E_1,E_2$$ and the fact that the slope of the middle term of an exact sequence if a convex combination of the slopes of the other two terms, one finds $$\mu(F)\le \mu$$.
About question 2, this is not exactly what Wells is saying even though this is not relevant for your question. Take a semistable bundle E that is not stable. Then, take $$F\subset E$$ the subbundle with smallest rank among the subbundles that have same slope as $$E$$. Then $$F$$ is automatically stable. Moreover, as the slope of $$E$$ is a convex combination of the slope of $$F$$ and $$E/F$$, those three quantities must coincide. Finally, $$E/F$$ is semistable. To see that, use the characterization of semistability using quotient bundles instead of subbundles. Any quotient of $$E/F$$ is again a quotient of $$E$$. As $$E$$ and $$E/F$$ have the same slope, you are done.