TITLE: Sum of Cauchy sequences is Cauchy in an Abelian Topological Group
QUESTION [5 upvotes]: 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.

REPLY [10 votes]: You had the right idea, so let me spell out the argument with the fix I suggested in the comments:
Suppose that for every $0$-neighborhood $U$ there is a $0$-neighborhood $V$ such that $V + V \subset U$. Since $(x_n)$, and $(y_n)$ are Cauchy, there is $N$ such that $x_n - x_m \in V$ and $y_n - y_m \in V$ for all $m,n \geq N$ and hence $(x_n+y_n)-(x_m+y_m) = (x_n-x_m)+(y_n-y_m) \in V+V \subset U$ for all $m,n \geq N$, showing that $(x_n+y_n)$ is Cauchy.
To see that our hypothesis is in fact true, we can argue as follows: Since the addition map $a\colon G \times G \to G, (g,h) \mapsto g+h$ is continuous, we know that $a^{-1}(U) \subset G \times G$ is open. Also, $(0,0) \in a^{-1}(U)$, so there are open $0$-neighborhoods $V_1, V_2 \subset G$ such that $V_1 \times V_2 \subset a^{-1}(U)$ by the definition of the product topology. Now $V = V_1 \cap V_2$ is a $0$-neighborhood such that $V + V \subset U$, since $V \times V \subset V_1 \times V_2 \subset a^{-1}(U)$.