I'm reading chapter 10 of A-M in Completions, and I'm trying to understand how coherent sequences give rise to Cauchy sequences. This seems to be pretty clear, according to the text, but it eludes me.
Specifically, it says the following:
Given any coherent sequence $(\zeta_n)$, (in the sense that $\theta_{n+1}(\zeta_{n+1})=\zeta_n$, for all $n$, where $\theta_{n+1}:G/G_{n+1}\to G/G_n$) we can construct a Cauchy sequence $(x_n)$ giving rise to it, by taking $x_n$ to be any element in the coset of $\zeta_n$( so that $x_{n+1}-x_n \in G_n$).
I can't see why the very last part follows, namely $x_{n+1}-x_n \in G_n$?
Thanks!
$\zeta_n$ is a left coset of $G_n$ in $G$, i.e. and element of $G/G_n$.
For each $n$, $x_n$ is chosen so that $x_n + G_n = \zeta_n$.
By definition of $\theta$ we have $\theta_{n+1} (x_{n+1} + G_{n+1}) = x_{n+1} + G_n$.
So $\theta_{n+1}(\zeta_{n+1})=\zeta_n$ gives $x_{n+1}+G_n = x_n + G_n.$
So $x_{n+1}-x_n \in G_n$.