Proposition 5

Let $A$ be a ring and $A$ and $R$ two (commutative)A-algebras. If $R$ is integral over $A$, $R \otimes_A A'$ is integral over $A'$.

Consider any element $x' = \sum_{i=1}^n x_i \otimes a'_i$ of $R \otimes_A A'$, where the $x_i$ belong to $R$ and the $a'_i$ to $A'$; as $x_i \otimes a'_i = (x_i \otimes 1)a'_i$ and the $x_i \otimes 1$ are integral over $A'$ (Proposition 2), so is $x$.