I was reading exercise 9.5 of Matsumura's Commutative Algebra. The problem goes like this:
Let $A$ be an integral domain and $K$ its field of fractions. We say that $x\in K$ is almost integral over $A$ if there exists $0\ne a\in A$ such that $ax^n\in A$ for all $n>0$. Say that $A$ is completely integrally closed if every $x\in K$ which is almost integral over $A$ belongs to $A$. Prove that if $A$ is completely integrally closed, then so is $A[[X]]$.
I have no idea how to use the condition that $A$ is completely integrally closed. The only possible solution I can think of is to compute directly, which seems to be tedious and difficult.
