Let $G_1,\cdots, G_n, \cdots$ metrizable topological groups. Prove that $G= \prod_{n\in \mathbb N} G_n$ is also a topological group.
My book does not define the operations that we should have in $G$. So I will state them (as they naturally should be) and give an attempt to show that they are continuous.
Define $\mu: G\times G \rightarrow G$ given by $\{(a_i)_{i\in \mathbb N}, (b_i)_{i\in \mathbb N}\} \mapsto (a_i\cdot b_i)_{i\in \mathbb N}$ and $\phi: G\rightarrow G$ given by $(a_i)_{i\in \mathbb N} \mapsto (a_i^{-1})_{i\in \mathbb N}$.
I will only prove that $\mu$ is continuous (to the other function the argument is similar). It is then sufficient to show that for every $j\in \mathbb N$ the function $\mu_i = p_i \circ \mu$ is continuous. Hence, given $j$ we see that $\mu_j((a_i)_{i\in \mathbb N}, (b_i)_{i\in \mathbb N}) = a_j\cdot b_j = m(a_j,b_j)$, where $m: G_i \times G_j\rightarrow G_j$. Since $G_j$ is a topological group, we have (by definition) that $m$ is continuous, which shows that $\mu_j$ is also continuous. Hence comes the continuity of $\mu$.
Is this proof fine? Am i missing something?
