Cfrgtkky

Following Atiyah MacDonald (Chapter 10: Completions), let $G$ be a topological abelian group. We assume that $0 in G$ has a fundamental system of neighborhoods consisting of subgroups $G = G_0 supseteq G_1 supseteq … supseteq G_n supseteq … $
Suppose $(x_k)$ is a Cauchy sequence in $G$.
Then it is claimed, that the image of $x_k$ in $G/G_n$ is ultimately constant. Why is this the case?

Saying that $(x_n)$ is Cauchy means that, for every $k$, there exists $N$ such that, for $m,n>N$, $x_m-x_nin G_k$.

Thus, for $m,n>N$, $x_m+G_k=x_n+G_k$. This is the same as saying that the image of $(x_n)$ in $G/G_k$ is eventually constant.

Definition: A Cauchy sequence in $G$ is a sequence $(x_k)$ of elements of $G$ such that, for any neighborhood $U$ of $0$, there exists an integer $s(U)$ with the property that $x_m-x_l in U$ for all $m,l geq s(U)$.

My idea:
Let $phi: G rightarrow G/G_n$ denote the projection sending $x_k$ to $x_k +G_n$.

Choose $U = G_n$ $Rightarrow x_m-x_l in G_n$ for all $m,l geq s(G_n)$

$Rightarrow x_m+G_n = x_l +G_n$ for all $m,l geq s(G_n)$

$Rightarrow phi(x_m) = phi(x_l)$ for all $m,l geq s(G_n)$

$Rightarrow$ For all $r geq s(G_n)$ we have $phi(x_r) = lambda_n$