The Stacks project

Proof. Let $I_\lambda$ be a fundamental system of open ideals of $A$. Recall that $A^\wedge = \mathop{\mathrm{lim}}\nolimits A/I_\lambda$ with the limit topology, which means that the kernels $J_\lambda = \mathop{\mathrm{Ker}}(A^\wedge \to A/I_\lambda )$ form a fundamental system of open ideals of $A^\wedge$. Since $J_\lambda$ is the closure of $I_\lambda A^\wedge$ (compare with Lemma 87.4.11) we conclude. $\square$