I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: \mathbb{P}(\mathcal{E}) \to X$ is projective bundle associated to locally free sheaf $\mathcal{E}$ on $X$).
If $\mathcal{E}$ is gbgs, we can pullback the surjection $\mathcal{O}_X^n \to \mathcal{E}$ to obtain a surjection $\mathcal{O}_{\mathbb{P}(\mathcal{E})}^n \to \pi^* \mathcal{E}$ (as pullback is right exact), which we can then compose with natural surjection $\pi^* \mathcal{E} \to \mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$.
I have trouble with opposite direction, though. Suppose we have a surjection $\mathcal{O}_{\mathbb{P}(\mathcal{E})}^n \to \mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$. We can pushforward it via $\pi$ to obtain a map $\pi_* \mathcal{O}_{\mathbb{P}(\mathcal{E})}^n \to \pi_*\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$, but since $\pi_* \mathcal{O}_{\mathbb{P}(\mathcal{E})}^n \simeq \mathcal{O}_X^n, \pi_*\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1) \simeq \mathcal{E}$, we have a map $\mathcal{O}_X^n \to \mathcal{E}$. As pushforward is only left exact, we cannot really conclude that it will be surjective as well. Also, $\pi$ is not affine, so higher direct images don't have to vanish.
Any hints?