Vector bundle on projective curve/Strong Harder-Narasimhan filtration/Existence/Fact

Let ${}C$ denote a smooth projective curve over an algebraically closed field of positive characteristic ${}p$, and let ${}{\mathcal {S}}$ be a vector bundle on ${}C$.

Then there exists a natural number

${}e\in \mathbb {N}$ such that the Harder-Narasimhan filtration of the ${}e$th Frobenius pull-back ${}F^{e*}({\mathcal {S}})$, say

${}0={\mathcal {S}}_{0}\subset {\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t-1}\subset {\mathcal {S}}_{t}=F^{e*}({\mathcal {S}})\,$

has the property that the quotients ${}{\mathcal {S}}_{k}/{\mathcal {S}}_{k-1}$ are strongly semistable.

Write a proof