Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is *ample* if the relative twisting sheaf $\mathcal{O}\_{\mathbf{P}(E)}(1)$ is an ample line bundle on the projective bundle $\mathbf{P}(E)$ of hyperplanes in $E$. Now let $E$ be an arbitrary vector bundle and let $L$ be an ample line bundle on $X$. Will $E \otimes L^{\otimes n}$ be an ample vector bundle for $n \gg 0$? Of course it would suffice if the pullback of $L$ to $\mathbf{P}(E)$ was still ample. But in general pullbacks of ample line bundles won't remain ample.



Making a vector bundle ample by twisting with ample line bundle
Because $L$ is ample, $E\otimes L^n$ is generated by global sections for $n\gg 0$, i.e., there is a surjective morphism $\mathscr O\_X^{\oplus r} \to E\otimes L^n$, which implies that there is a surjective morphism $L^{\oplus r} \to E\otimes L^{n+1}$. As $L$ is ample, so is $L^{\oplus r}$ and then so is its quotient $E\otimes L^{n+1}$.


