I was reading exercise 27, Chapter V of Bourbaki's Commutative Algebra. The problem goes like this;
Let $R$ be an integrally closed domain that is not completely integrally closed, $X$ an indeterminate of $R$ and $a$ a nonzero nonunit of $R$ such that $\bigcap_{i=1}^{\infty}a^{i}R\neq 0$. Show that there exists an element $f=\sum_{n=1}^{\infty}u_{n}X^{n}$ of the quotient field of $R[[X]]$ such that $f^{2}-af+X=0$ but $f\not\in R[[X]]$, so $R[[X]]$ is not integrally closed.
My question is, can the assumption that $R$ is integrally closed be dropped? I do understand that we have to assume that $R$ is not completely integrally closed since such $a$ would not exist otherwise, but I cannot find where $R$ being integrally closed is needed in this proof.