We know that, given a first countable abelian topological group $G$, the sum of two Cauchy sequences gives yet another Cauchy sequence (see, e.g., this answer).
For those wondering, we say that a sequence $(x_n)$ in $G$ is Cauchy if for any neighborhood $U$ of $0$, there exists an integer $s = s(U)$ such that $x_n - x_m \in U$ whenever $n,m \geq s$.
However, I don't see where the countability hypothesis is used. I know that it is required, as there exist counterexamples (take $G = \mathbb{R}$ with the metric $d(x,y) = |\arctan(x)-\arctan(y)|$ and the Cauchy sequences $(x_n),(y_n)$ with $x_n = n, \, y_n = (-1)^n - n$) (This is actually first-countable, as a metric space).
Update: I am essentially wondering why Atiyah-Macdonald restrict themselves to groups where $0$ has a countable neighborhood basis when talking about these concepts. Let's include a picture:
Can anyone enlighten me?
Thank you in advance.
No, first countable is not needed for the Cauchyness of sum of Cauchy sequences, the given proof uses only that for each open neighborhood $U$ there is a $V$ such that $V+V\subseteq U$, which is basically the continuity of addition.
Instead, your counterexample is rather tricky! They are Cauchy sequences with respect to the given metric, but they are not Cauchy as defined from the topological group structure (I guess the usual addition is used)..
Anyway, $\Bbb R$ is first countable.