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Let $X$ be a smooth projective curve over $\mathbb{C}$. Then I want to show that for any coherent sheaf $\mathcal{F}$ on $X$ there exists an integer $d$ such that for any subsheaf $\mathcal{F}'\subset \mathcal{F}$ we have $$ \operatorname{deg}(\mathcal{F}') \leq d. $$ Is there a reference for that?

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    $\begingroup$ What definition of degree are you using? $\endgroup$ – Mohan Sep 17 '15 at 16:19
  • $\begingroup$ for a line bundle $\mathcal{L}\cong \mathcal{O}_X(\sum a_i P_i)$ the degree is $\sum a_i$, for higher rank bundles it is determined by additivity of short exact sequences $\endgroup$ – Jimmy R Sep 17 '15 at 16:37
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    $\begingroup$ This definition makes sense only for curves. $\endgroup$ – Mohan Sep 17 '15 at 20:19
  • $\begingroup$ sorry, you're right! I've edited my question accordingly $\endgroup$ – Jimmy R Sep 17 '15 at 20:25
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For your question, you may assume that $F$ is a vector bundle (why?) and then it has a filtration by line bundles, say $L_1,L_2,\ldots L_n$. Choose $d$ to be larger than sum of the absolute values of degrees of all the $L_i$s.

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