I am trying to understand the proof of the existence of Harder-Narasimhan filtration from Huybrechts and Lehn.
Let $X$ be a projective scheme with a fixed ample line bundle. Then the theorem says that every pure sheaf has a unique Harder-Narasimhan filtration.
The book first proves the following lemma : let $E$ be a purely $d$-dimensional sheaf. Then there is a subsheaf $F\subset E$ such that for all subsheaves $G\subset E$ one has $p(F)\geq p(G) $ and in case of equality $G\subset F$. Moreover $F$ is uniquely determined and semistable. $F$ is the maximal destabilizing subsheaf.
My doubt is as follows. Once we establish the existence of such an $F$, the book says by induction we can assume that $E/F$ has a Harder Narasimhan filtration.
What are we inducting on? My guess is the dimension of the sheaf $E$. But if so I am not able to see why dimension of $E/F$ is strictly less than dimension of $ E$. Any help will be appreciated!
I think we can induct on the rank of $E$. Look at the sequence $0 \to F \to E \to E/F \to 0$. Since $rk(E) = rk(F) + rk(E/F)$, and $F$ is a proper nonzero subsheaf of $E$, so $rk(E/F) < rk(E)$, and so we can proceed via induction.