How to find a sequence in discrete group

Question

Let $\Gamma$ be a discrete group. Can we find an increasing sequence $F_{n}\subset \Gamma$ of finite subsets, such that $\cup F_{n}=\Gamma$?

Answer 1

Your group needs to be countable- as countable union of finite sets is countable.

Now, if $\Gamma$ is countable, then enumerate all elements of $\Gamma$ as $\{g_1,g_2,\cdots,g_n,\cdots\}$, and take $F_1=\{g_1\}$, .. $F_n=\{g_1,g_2,\cdots,g_n\}$.