Lemma 87.7.6. Let $B \to A$ and $B \to C$ be continuous homomorphisms of linearly topologized rings. If $A$ and $C$ are weakly pre-adic, then $A \widehat{\otimes }_ B C$ is weakly adic.

Proof. We will use the characterization of Lemma 87.7.2 without further mention. By Lemma 87.4.12 we know that $A \widehat{\otimes }_ B C$ is admissible. Moreover, the proof of that lemma shows that the closure $K \subset A \widehat{\otimes }_ B C$ is an ideal of definition, when $I \subset A$ and $J \subset C$ of $I(A \widehat{\otimes }_ B C) + J(A \widehat{\otimes }_ B C)$ are ideals of definition. Then it suffices to show that the closure of $K^ n$ is open for all $n \geq 1$. Since the ideal $K^ n$ contains $I^ n(A \widehat{\otimes }_ B C) + J^ n(A \widehat{\otimes }_ B C)$, since the closure of $I^ n$ in $A$ is open, and since the closure of $J^ n$ in $C$ is open, we see that the closure of $K^ n$ is open in $A \widehat{\otimes }_ B C$. $\square$

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