Is there a direct proof that $\mathcal O(n) \in \text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\} $ for all $n\in \mathbb Z$

Question

Let $k$ be an algebraically closed field, and let $\mathbb P^1$ denote $\mathbb P^1_k$. Let $D^b(\mathbb P^1)$ be the bounded Derived Category of Coherent Sheaves on $\mathbb P^1$. Let $\text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\}$ be the intersection of all thick subcategories ( https://ncatlab.org/nlab/show/thick+subcategory) of $D^b(\mathbb P^1)$ containing $\mathcal O$ and $\mathcal O(1)$.

Is there a direct proof (without going into semiorthogonal decomposition) of the fact that for every $n\in \mathbb Z$, we have $\mathcal O(n) \in \text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\} $ ?

Answer 1

For any $n > 0$ there are exact sequences $$ 0 \to \mathcal{O}^{\oplus (n-1)} \to \mathcal{O}(1)^{\oplus n} \to \mathcal{O}(n) \to 0 $$ and $$ 0 \to \mathcal{O}(-n) \to \mathcal{O}^{\oplus (n+1)} \to \mathcal{O}(1)^{\oplus n} \to 0. $$