Fix a proof

Thanks to Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/0C0D#comment-8076

more-morphisms.tex

@@ -7199,13 +7199,15 @@ \section{Reduced fibres}
 \end{lemma}
 
 \begin{proof}
-Let $x \in X$ be a point in the generic fibre $X_\eta$
-such that $\mathcal{O}_{X_\eta}$ is nonreduced.
-Then $\mathcal{O}_{X, x}$ is nonreduced.
+Assume the special fibre is reduced.
+Let $x \in X$ be any point, and let us show that $\mathcal{O}_{X, x}$
+is reduced; this will prove that $X$ and $X_\eta$ are reduced.
 Let $x \leadsto x'$ be a specialization with $x'$
 in the special fibre; such a specialization exists
 as a proper morphism is closed. Consider the local
-ring $A = \mathcal{O}_{X, x'}$. Let $\pi \in R$ be a uniformizer.
+ring $A = \mathcal{O}_{X, x'}$. Then $\mathcal{O}_{X, x}$
+is a localization of $A$, so it suffices to show that
+$A$ is reduced. Let $\pi \in R$ be a uniformizer.
 If $a \in A$ then there exists an $n \geq 0$ and an element
 $a' \in A$ such that $a = \pi^n a'$ and $a' \not \in \pi A$.
 This follows from Krull intersection theorem