# Statement of Lech's lemma

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.

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 $i$), $$\left|\frac{l(M/(x_1^{\nu_1},\ldots ,x_d^{\nu_d})M)}{\nu_1\cdots \nu_d}-e(\mathfrak{q},M)\right|<\epsilon.$$