A question on divisible group

Question

Let $G$ be a divisible, locally compact abelian group and $L_{1}\supseteq L_{2}\supseteq L_{3}\supseteq ...$ be a sequence of compactly generated, open subgroups of $G$. Can we deduce that there exist $n$ such that $L_{i}=L_{n}$ for $i\geq n$?

Answer 1

Not without additional assumptions.

Let $G=\mathbb{Q}$ with the discrete topology, or $(\mathbb{R}, +)$ with the usual topology. Let $L_n=2^n\mathbb{Z}$.

As far as what additional assumptions might make this true, I can't think of any off the top of my head. Perhaps if you gave some motivation I might be able to steer you in the right direction.