Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$. Is the following claim true?

Claim: For any $\epsilon>0$, there exists a positive integer $n$ s.t. for any ideal $I$ satisfying

1) $ I \subset\mathfrak{m^n}$

2) $\sqrt I = \mathfrak{m}$

3) $I$ can be generated by $d$ elements,

the following holds: $$ \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$

Note: The following example shows that the claim can be false if one drops the requirement that that the number of generators of $I$ be bounded.

Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,s)$. Let $I_{n,m}$ be an ideal of $A$ given by $$ I_{n,m}= s\mathfrak{m}^{n-1} + \mathfrak{m}^m$$.

We can calculate that for any $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$