I'm trying to solve an exercise regarding sequences on topological groups, Consider $G$ a topological group, i have to prove that the class of convergent ( to $0$ ) sequences is closed in the class of Cauchy sequences with respect to the induced topology by the infinite product $\prod_{n} G$.
My idea was to prove that the complementary is open, take $ \{ x_n \}$ non convergent to zero, exist an open NGH $U$ of $0$, s.t. $x_n \not\in U$ for $n> N_U$. Then my idea was that exist an open NGH $V$ of $x_n$ disjoint with $U$ and so take the inverse image of the canonical projection of $V$ in the product.
But i can't prove that this $V$ exists.
Some suggestions? Thank you .