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Since $r^n+a_1r^{n-1}+...+a_n=0$ is an equation of integral dependence of $r$ over an ideal $I$, by definition, $a_i\in I^i$. So
$a_1\in I$ is generated by one element of $I$. till now nothing goes wrong if we consider the ideal $(a_1)$ instead of $I$.
$a_2\in I^2$; let $a_2 $ is sum of products like $t_{k1} t_{l2}$ (t_k1 ,t_l2 are elements of I). so $a_2$ is generated by finite elements of $I$. till now, nothing goes wrong if we consider the ideal $(a_1,\{t_{k1}\}_k,\{t_{l2}\}_l)$ instead of $I$
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write the same argument for $a_i$. and consider the ideal generated by these finite elements