Lemma 10.59.2. Suppose that $M' \subset M$ are finite $R$-modules with finite length quotient. Then there exists a constants $c_1, c_2$ such that for all $n \geq c_2$ we have
\[ c_1 + \chi _{I, M'}(n - c_2) \leq \chi _{I, M}(n) \leq c_1 + \chi _{I, M'}(n) \]
Proof.
Since $M/M'$ has finite length there is a $c_2 \geq 0$ such that $I^{c_2}M \subset M'$. Let $c_1 = \text{length}_ R(M/M')$. For $n \geq c_2$ we have
\begin{eqnarray*} \chi _{I, M}(n) & = & \text{length}_ R(M/I^{n + 1}M) \\ & = & c_1 + \text{length}_ R(M'/I^{n + 1}M) \\ & \leq & c_1 + \text{length}_ R(M'/I^{n + 1}M') \\ & = & c_1 + \chi _{I, M'}(n) \end{eqnarray*}
On the other hand, since $I^{c_2}M \subset M'$, we have $I^ nM \subset I^{n - c_2}M'$ for $n \geq c_2$. Thus for $n \geq c_2$ we get
\begin{eqnarray*} \chi _{I, M}(n) & = & \text{length}_ R(M/I^{n + 1}M) \\ & = & c_1 + \text{length}_ R(M'/I^{n + 1}M) \\ & \geq & c_1 + \text{length}_ R(M'/I^{n + 1 - c_2}M') \\ & = & c_1 + \chi _{I, M'}(n - c_2) \end{eqnarray*}
which finishes the proof.
$\square$
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