Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

Question

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you have a great reference with a proof? I have next to no knowledge on Lie groups at all, but I do need the result to prove that non-solvable subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are Powers groups (in an operator algebra context).