The Stacks project

Proof. We may assume $r_ i \geq 1$ for all $i$. We will prove the lemma by induction on $\sum (r_ i - 1)$. If this integer is $0$, then $\mathcal{E}_ i$ is invertible for all $i$ and we conclude by taking $\pi = \text{id}_ X$. If not, then we can pick an $i$ such that $r_ i > 1$ and consider the projective bundle $p : P \to X$ associated to $\mathcal{E}_ i$. We have a short exact sequence

of finite locally free $\mathcal{O}_ P$-modules of ranks $r_ i - 1$, $r_ i$, and $1$. Observe that $p^* : A(X) \to A(P)$ is injective by assumption. By the induction hypothesis applied to the finite locally free $\mathcal{O}_ P$-modules $\mathcal{F}$ and $p^*\mathcal{E}_{i'}$ for $i' \not= i$, we find a morphism $p' : P' \to P$ with properties stated as in the lemma. Then the composition $p \circ p' : P' \to X$ does the job. $\square$