Why is any rank $1$ sheaf always stable?

Question

Let $X$ be a projective scheme over $\mathbb{C}$. For a sheaf on $X$, $$ p_E(d)=\chi(X,E(d)) $$ be the Hilbert polynomial of $E$. A sheaf $E$ on $X$ is siad to be stable if for every proper subsheaf $F\subset E$, $$ p_F(d)/rk(F)<p_E(d)/rk(E) $$ for sufficiently large $d>0$. Why is any rank $1$ sheaf always stable?

Answer 1

If $F$ is a non-zero sub-coherent sheaf of $E$, then it also has rank 1. Let $S=E/F$. Then $p_E(d)=p_F(d)+p_S(d)$ with $p_S(d)>0$ when $d$ is big enough. Thus $$p_F(d)/\mathrm{rk}(F)=p_F(d) < p_E(d)=p_E(d)/\mathrm{rk}(E).$$