I want to describe all Lie subgroups of the real line. I know that it is the same as to describe all closed subgroups and I know how to do that for $S^1$. Next I wanted to consider $$\pi: \mathbb{R} \to \mathbb{R/Z} = S^1 $$ and look at $\pi^{-1}(subgruops)$ but a problem occurs: I need this map to be closed (take closed subsets to closed) but it is not.
Can I modify this proof to make it work or do I need something different?( I know that there are different approaches but I want to fix mine)
No, it won't work: the group $\sqrt{2}\mathbb Z$ is a Lie subgroup of $\mathbb R$, but under the mapping $\pi: \mathbb R \to \mathbb R / \mathbb Z = S^1$ it does not get taken to a closed subgroup of $S^1$.