I'm reading the section of Le Potier's 'Lectures on Vector Bundles' where he proves that the Grothendieck group $K(X)$ of a smooth complex projective curve $X$ (which is the free abelian group on coherent sheaves on $X$ modulo the relations $F = F' + F''$ for exact sequences $0 \to F' \to F \to F'' \to 0$) is in fact equal to $A(X) \oplus \mathbb{Z}$ where $A(X)$ is the subgroup generated by the structure sheaves of divisors (things like $\mathcal{O}_{X,P}/\mathfrak{m}_P^n$) and $\mathbb{Z}$ corresponds to the trivial bundles of all ranks.
The main task is to show that every vector bundle $F$ can be written as an element of $A(X) \oplus \mathbb{Z}$. If $F$ were generated by its sections, then we can consider an appropriate trivial bundle $\mathcal{O}^r$ such that $\mathcal{O}^r \to F$ is an isomorphism on an open set and so, we'll have - $$0 \to \mathcal{O}^r \to F \to \Theta \to 0$$ where $\Theta$ is going to be the structure sheaf of a divisor. Thus $F \in A(X) \oplus \mathbb{Z}$. Now, for a general $F$, some high twist $F(n)$ is going to be generated by its sections and so, $F(n) \in A(X) \oplus \mathbb{Z}$. Then, he says "It is clear that every element of $A(X)$ has square zero in $K(X)$ ($K(X)$ has a ring structure given by tensor product). Therefore, $\mathcal{O}(-n)$ is contained in this subgroup and this subgroup is in fact a subring. It follows that $K(X)=A(X) \oplus \mathbb{Z}$".
To prove that $O(-n)$ is in this subgroup, I considered the short exact sequence $$0 \to \mathcal{O}(-n) \to \mathcal{O} \to \mathcal{O}_{X,P}/\mathfrak{m}_{P}^n \to 0$$ which implies that $\mathcal{O}(-n) \in A(X) \oplus \mathbb{Z}$. Now, for a general $F$, if I were to take the short exact sequence for some suitable $n$ - $$ 0 \to \mathcal{O}^r \to F(n) \to \Theta \to 0$$ and then twist this by $\mathcal{O}(-n)$ to get - $$0 \to \mathcal{O}(-n)^r \to F \to \Theta' \to 0$$ $F$ would be in $A(X) \oplus \mathbb{Z}$ as $\mathcal{O}(-n)$ and $\Theta'$ both are.
I am not able to see the significance of the quoted paragraph and why it is relevant to the proof, I would be extremely grateful if somebody could point that out. I am also not entirely sure whether I am missing something in my proof. Thanks in advance!