Integral closure of an ideal $I$ of a ring $A$ is defined as follows: $$I^{int}:=\{x\in A~ |~\exists~ n\in \mathbb{N}~\text{and}~a_i\in I^i~\text{for}~1\leq i\leq n~ ~\text{such that}~ x^n+\sum_{i=1}^{n}a_ix^{n-i}=0~\}$$ Since it is a closure operation, it must satisfy the idempotence condition i.e., $(I^{int})^{int}$= $I^{int}$. However, I am not being able to show that $(I^{int})^{int} \subseteq I^{int}$. Can someone please help me out?
Thanks in advance!