A group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ? Is it true that a group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?

As per the request in the comments, here is the proof in the case that $G$ is countably generated.
We choose a countable generating set $\{a_n~|~n \in \mathbb{N}\}$ of $G$ and define for $N \in \mathbb{N}$ the subgroup $C_N:=\langle a_1,...,a_N \rangle$. 
Then $C_N$ is finitely generated hence cyclic and the set of all the $C_N$ forms a chain by construction. Moreover $\bigcup_N C_N$ contains a generating system for $G$ and thus coincides with $G$ which proves the assertion.
