No non-trivial homomorphism to a group

Question

Let $G$ be a compact Hausdorff topological group, and let $H$ be a torsion-free group satisfying the ascending condition, i.e. there are no infinite strictly ascending chains $H_1<H_2<...$ of subgroups of $H.$

Prove that there is no non-trivial homomorphism of $G$ into $H.$

Note: no topology is considered on $H$ and "homomorphism" simply means "group homomorphism."

Answer 1

It is easy to see that such an $H$ is finitely generated and the rest follows from Nikolov-Segal theorem.

I wonder if there is a non high-tech way to prove it though!