Let $X$ be a smooth connected projective curve of genus $g$ over an algebraically closed field. Let $E$ be a vector bundle on $X$ of rank $n$. Is it true that there exists a constand $N(g,n)$ such that each such $E$ has a filtration $$ 0\subset E_1\subset ...\subset E_n=E $$ by subbundles with $rank(E_i)=i$, such that $deg(E_i/E_{i1})deg(E_{i+1}/E_i)\geq N(g,n)$ ? What is the reference for this?

According to lemma 4 of M.F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414452. you could take $N(g,n) =  2 g$. 

