Let $G$ be a connected Lie group of dimension one and let $U=\{u_t\}_{t \in \mathbb R }$ be a one-parameter subgroup. I wonder if it is true that $U=G$?
It is easy to see that $U$ is closed. So by a theorem in Lie theory, $U$ is a Lie subgroup. $U$ is not discrete and thus it must have dimension one. But I don't know how to go further
Since $\dim U=\dim G$, $U$ is open in $G$, and thus clopen. Since $G$ is connected, this implies $U=G$.