Suppose $(G_n, p_{n+1,n}:G_{n+1}\to G_n)$ is an inverse sequence of discrete groups and (obviously continuous) group homomorphisms. Let $G=\varprojlim G_n$ be the inverse limit (with the usual inverse limit topology) and $H\leq G$ be a finitely generated subgroup.
Question: Must $H$ be closed in $G$?
The answer is negative if $H$ is allowed to have infinitely many generators, however, it seems plausible that $H$ is a discrete subgroup of $G$ if it is finitely generated and this would do the trick.
I am mainly interested in the case where $G_n=F(x_1,...,x_n)$ is the free group on n-generators and $$p_{n+1,n}(x_{j})=\begin{cases} x_j, & 1\leq j\leq n\\ 1, & j=n+1 \end{cases}$$ In this case, $H\leq G$ is free whenever it is finitely generated.
The answer is negative. Consider for instance the group $G$ of p-adic integers. This groups is inverse limit of finite cyclic groups and is compact. Now, let $H$ be the subgroup of the usual integers in $G$.