Lemma 87.5.4. Let $A \to B$ and $B \to C$ be continuous homomorphisms of linearly topologized rings. If $A \to B$ and $B \to C$ are taut, then $A \to C$ is taut.

**Proof.**
Omitted. Hint: if $I \subset A$ is an ideal and $J$ is the closure of $IB$, then the closure of $JC$ is equal to the closure of $IC$.
$\square$

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