The Stacks project

Proof. Recall that a morphism of schemes $V \to W$ is a monomorphism if and only if the diagonal $V \to V \times _ W V$ is an isomorphism (Schemes, Lemma 26.23.2). The morphism $X_0 \to X_0 \times _{Y_0} X_0$ is locally of finite presentation by Morphisms, Lemma 29.21.12. Since $X_0 \times _{Y_0} X_0$ is quasi-compact and quasi-separated (Schemes, Remark 26.21.18) we conclude from Lemma 32.8.11 that $\Delta _ i : X_ i \to X_ i \times _{Y_ i} X_ i$ is an isomorphism for some $i \geq 0$. For this $i$ the morphism $f_ i$ is a monomorphism. $\square$