Lemma 87.7.5. Let $\varphi : A \to B$ be a continuous homomorphism of linearly topologized rings. If $\varphi $ is taut and $A$ is weakly pre-adic, then $B$ is weakly pre-adic.
Proof. Let $I \subset A$ be an ideal such that the closure $I_ n$ of $I^ n$ is open and these closures define a fundamental system of open ideals. Then the closure of $I^ nB$ is equal to the closure of $I_ nB$. Since $\varphi $ is taut, these closures are open and form a fundamental system of open ideals of $B$. Hence $B$ is weakly pre-adic. $\square$
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