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 $deg(E\_i/E\_{i-1})-deg(E\_{i+1}/E\_i)\geq N(g,n)$ ?
What is the reference for this?



FIltrations on a vector bundle on a curve
According to lemma 4 of 


M.F. Atiyah, *Vector bundles over an elliptic curve*, Proc. London Math. Soc. (3) 7 (1957), 414-452.


you could take $N(g,n) = - 2 g$. 


