Let $H$ be a subgroup of $\Bbb R^*$. If $\Bbb R^+ \subseteq H \subseteq \Bbb R^*$, prove that $H = \Bbb R^+$ or $H = \Bbb R^*$. 
Let $H$ be a subgroup of $\Bbb R^*$, the group of nonzero real numbers under multiplication. If $\Bbb R^+ \subseteq H \subseteq \Bbb R^*$, prove that $H = \Bbb R^+$ or $H = \Bbb R^*$.

I think that $\Bbb R^+$ is supposed to mean the set of non-negative reals? I am a bit confused on what do I need to prove here? If I assume that $H \ne \Bbb R^+$ and get that $H=\Bbb R^*$, then would that prove the result?
Suppose that $H \ne \Bbb R^+$, then I only have that there exists $h \in H$ such that $h \notin \Bbb R^+$?

Consider the canonical submersion $\pi:\Bbb R^*\to \Bbb R^*/\Bbb R^+=\Bbb Z_2$.  By the correspondence theorem, there's a bijection between the possible  subgroups $H$ such that $\Bbb R^+\le H\le\Bbb R^*$ and the subgroups of $\Bbb Z_2$.  Hence there are only two possibilities.
