# Index of subgroups in residually finite groups

Let $$G$$ be an infinite finitely generated residually finite group. Is it true that $$G$$ contains finite-index subgroups of arbitrarily large index?

What about the converse: does there exist a finitely generated group with finite-index subgroups of arbitrarily large index that is not resitually finite? Is there any example of such a group that is furthermore amenable?

Nope. There's a simpler argument, not relying on finite generation, that in every infinite residually finite group the subgroups of finite index have arbitrarily large index. Let $$G$$ be such a group and let $$g_1 \in G$$ be nonzero. By residual finiteness it avoids some finite index subgroup $$H_1$$. Now let $$g_2 \in H_1$$ be nonzero. By residual finiteness it avoids some finite index subgroup $$H_2$$. The intersection $$H_1 \cap H_2$$ is also finite index and has index strictly larger than $$H_1$$. Now let $$g_3 \in H_1 \cap H_2$$ be nonzero, etc. Continuing in this way we build a sequence $$H_i$$ of subgroups of strictly increasing index. (And we know that each $$H_i$$ has a nonzero element because $$G$$ is infinite.)
Here is an example as above: it is $$(\bigoplus\limits_{i \in \mathbb{Z}} A_5) \rtimes \mathbb{Z}$$, which is amenable, finitely generated, non-residually finite and contains subgroups of arbitrarily large finite index. I can provide the details if someone is interested.