Let $G$ be a topological abelian group and suppose $0$ has a countable fundamental system of neighborhoods. Let $(x_n),(y_n)$ be Cauchy sequences of $G$. Why is it true that $(x_n+y_n)$ is a Cauchy sequence?
I tried to generalize the case of real sequences: my problem is that if $U$ is a neighborhood of $0$, then i would need to use something like $\frac{1}{2} U$, but obviously this does not make sense.
I also looked at this relevant question Sum of Cauchy Sequences Cauchy? however it was not very helpful, since it refers to metric topological groups.
Thanks.