Is there a profinite group $G$ with a locally finite subgroup $H$ such that $\overline H$, the closure of $H$, is not torsion?



Looking for an example of profinite groups
You can take $G=\prod\_{k>0}\mathbb{Z}/2^k$ and $H=\bigoplus\_{k>0}\mathbb{Z}/2^k$. Then every finitely generated subgroup of $H$ is finite, and $\overline{H}=G$, but the element $(1,1,1,\dotsc)\in G$ is not torsion.


