Structure of countable infinite abelian torsion groups

Question

Let $G$ be a countable infinite abelian torsion group. How swhow that there exists a family $(G_n)_{n \geq 0}$ of finite subgroups such that $G_n \subsetneqq G_{n+1}$ for any $n \geq 0$ with $\cup_{n \geq 0} G_n =G$.

Do you know an explanation or a reference ?

Answer 1

Just enumerate $G$ as $\{g_k\}_{k\in\mathbb{N}}$ and let $G_n$ be the subgroup generated by $g_k$ for all $k\leq n$. This won't quite fit your requirements since you may have $G_n=G_{n+1}$, but to fix that you can just skip the duplicates in the sequence $(G_n)$.