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I'm self-learning Integral Closure of Ideal via the book of I.Swanson &C.Huneke.

In the begining of this book they say that:

A next attempt can be an asymptotic version. Let $v_n(r)$ be the least power of $r$ in $I^n$. If the limit of $\frac {v_n(r)}{n}$ exists and is at least 1, we can intuitively think that $r$ "grows" at least as fast as $I$.

Also, they say that this notion can be subsumed into a single equational definition:

Let $I$ be an ideal in a ring $R$. An element $r\in R$ is said to be integral over $I$ if there exist an integer $n$ and elements $a_i\in I^i$, $i=1,...,n$ such that

$r^n+a_1r^{n-1}+ a_2r^{n-2}+...+a_n=0$

Can anyone explain to me why the two notions are related ?

Anh_Rose 1210
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    Let me give you one way implication. If the equation you write is satisfied, I claim $r^{n+m}\in I^{m+1}$ for any $m\geq 0$ and $m=0$ follows from the equation. Multiply by $r^m$ the equation and use induction. So, we get $\nu_l(r)\leq l+n$ and thus the limit statement follows. – Mohan Aug 13 '17 at 01:59

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