Exercise 26, § 3

Let $B$ be a commutative ring and $A$ a complete Noetherian semi-local subring of $B$. Let $n$ be an ideal of $B$ containing a power of the Jacobson radical of $A$ and such that the $n$-adic topology on $B$ is Hausdorff. Then show that the topology on $A$ is induced by the $n$-adic topology on $B$ (use Proposition 8 of § 2, no. 7).