Show that every subgroup of $ ( \Bbb{R} ,+)$ is cyclic or dense.
(Note that we say $A \subset \Bbb{R} $ is dense if $\overline{A} =\Bbb{R}$.)
For example $\Bbb{Q}$ is a subgroup that is dense but not cyclic and $\Bbb{Z}$ is cyclic but not dense.
I think no subgroup of the real numbers can be both cyclic and dense.
