Showing that the radical contains the original ideal

Question

Let $I$ be an ideal in a commutative ring $R$ and let $$ J = \{ r \in R \mid \text{$r^n \in I$ for some positive integer $n$}\}. $$ Prove that $J$ is an ideal that contains $I$.

I can prove that $J$ is an ideal in $R$. But I don't know that $J$ contains $I$. Please guide me with a proof. Thank you for your kindness.

Answer 1

Certainly $J$ contains $I$. If $r \in I$, then $r^1 = r \in I$. So $r \in J$. Thus $I \subseteq J$.