The statement of the theorem $14.12$ (page 110) in Commutative ring theory by Matsumura is:
Let $A$ be a $d$-dimensional local ring, and $x_1,\ldots ,x_d$ be a system of parameters; set $\mathfrak q=(x_1,\ldots ,x_d)$, and suppose that $M$ is a finite $A$- module. Then $$e(\mathfrak q,M)=\lim_{\min (\nu_i)\rightarrow\infty}\frac{l(M/(x_1^{\nu_1},\ldots ,x_d^{\nu_d})M)}{\nu_1\cdots \nu_d}.$$
What I don't understand is what $\min (\nu_i)\rightarrow\infty$ means. Does it mean that all the $\nu_i$ goes to infinity seperately or only the minimum of the $\nu_i$ goes to infinity and other remains the same or something else.
Thank you in advance.
Well, if the minimum is going to infinity, then all the others have to go to infinity too, since the minimum is the smallest one!
To be perfectly precise, here's what that limit statement means. For any $\epsilon>0$, there exists an $N$ such that for any $\nu_1,\dots,\nu_d\in\mathbb{N}$ with $\min(\nu_i)\geq N$ (or equivalently, with $\nu_i\geq N$ for all $i$), $$\left|\frac{l(M/(x_1^{\nu_1},\ldots ,x_d^{\nu_d})M)}{\nu_1\cdots \nu_d}-e(\mathfrak{q},M)\right|<\epsilon.$$