The Stacks project

Proof. Let $J \subset B$ be a weak ideal of definition. Let $I \subset A$ be an open ideal such that $IB \subset J$. Let $I' \subset A$ be an open ideal. To show that $I$ is an ideal of definition we have to show that $I^ n \subset I'$ for $n \gg 0$. Denote $J' \subset B$ the closure of $I'B$. Then $A/I' \to B/J'$ is faithfully flat, hence injective. Thus in order to show that $I^ n \subset I'$ it suffices to show that $\varphi (I)^ n \subset J'$. This holds for $n \gg 0$ since $\varphi (I) \subset J$, the ideal $J$ is an ideal of definition of $B$, and $J'$ is open in $B$. $\square$