Remark

If the subgroup $H_\alpha$ is compact for some $\alpha \in I$, then $H_\beta$ is compact for all $\beta \geq \alpha$, because it is a closed subgroup of $H_\alpha$. Since the set of all $\beta \in I$ such that $\beta \geq \alpha$ is cofinal in $I$, it makes essentially no difference, in the study of the group $G$, whether we suppose that one of the $H_\alpha$ is compact or that all the $H_\alpha$ are compact.