Suppose that there is integral domain $I$. Now we take localization $I_m$ of $I$ respect to its maximal ideal $m$. $I_m$'s elements will consist of $a/b$ where $a \in I$ and $b \in m$.
But integral closedness is defined respect to field of fractions of $I_m$ and the field of fractions of $I_m$ is a subset of the field of fractions of $I$. Suppose that there exists elements not in $I_m$ but in field of fractions of $I$. Let those elements be $e=c/d$ where $c \in I$, $d \not \in m$. Then we can always multiply $e$ by $f/f$ where $f\in m$, which is same $e$, and $df \in m$. So it seems that it is impossible to find any element in the field of fraction of $I$ that is not in $I_m$. What did I do wrong here?