Cauchy sequence in a valuation ring

Question

From Janusz's book algebraic number fields, chapter 2.

Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal $\mathfrak p=R\pi$. Let $L$ be a finite, separable extension field of $K$ and $S$ the integral closure of $R$ in $L$ so $S$ is a free, f.g. $R$-algebra.

For any $R$-algebra $A$, Janusz defines a sequence $\{a_n\}$ to be Cauchy if there is a function $N(n.m)$ with integer values such that $a_n-a_m\in A\pi^{N(n,m)}$ where $N(n,m)\to \infty$ if $n,m\to \infty$.

Let $\{a_n\}$ be a Cauchy sequence in $S$. He writes $a_n=a^{(1)}_nx_1+...+a^{(m)}_nx_m$ where $x_1,...,x_m$ is an $R$-basis of $S$ and $a^{(i)}_n \in R$.

The book claims that for each $i$, $\{a_j^{(i)}\}$ is a Cauchy sequence in $R$ wrt $|\cdot|$. Why is this true? How can I show this for $m>1$?

Many thanks in advance .

Answer 1

Do you see why a sequence in $\mathbf{R}^N$ is Cauchy if and only if all its coordinate sequences are Cauchy in $\mathbf{R}$ ? Well, it's the same reason in your case.