In the article : On globally generated vector bundles on projective spaces, the authors prove the following lemma
Lemma 4. Let $\mathcal{E}$ be a generated vector bundle on $\mathbb{P}^{n}$ and $\mathcal{E}^{*}$ denote its dual bundle. If $h^{1}(\mathcal{E}^{*}) = s$ then there exists a globally generated vector bundle $\mathcal{G}$ such that $h^{0}(\mathcal{G}^{*}) = h^{0}(\mathcal{E}^{*}) $, $h^{1}(\mathcal{G}^{*}) = 0$ and an exact sequence $0 \longrightarrow \mathcal{O}_{\mathbb{P}^{n}}^{\oplus s} \longrightarrow \mathcal{G} \longrightarrow \mathcal{E} \longrightarrow 0 $.
In proof of this result, the authors state that the exact sequence above can be easily obtained by induction on s, having in mind that $\text{Ext}^{1}(\mathcal{E}, \mathcal{O}_{\mathbb{P}^{n}}) \simeq H^{1}(\mathcal{E}^{*})$. Is this really easy to obtain? Can someone help me?
Thank you.