If every quotient group of G by non-trivial normal subgroups are finite, then G is finite

Question

If every quotient group of $G$ by a non-trivial normal subgroup is finite, then $G$ is finite.

This a statement that I'm supposed to prove if it's true or not.

If $G = (\mathbb{Z}, +)$, then all non-trivial subgroups of $G$ are $n\mathbb{Z}$ with $n = 2,3,4,\dots\,$.

But as $G$ is abelian, so every subgroup is normal and the quotient group $\mathbb{Z}/n\mathbb{Z}$ is formed by $n$ elements. So this would be a counter-example to the previous statement. Is this a valid counterexample?

Answer 1

Yes. $\,\,\,\,\,\,\,\,\,\,\,\,$