I'm trying to solve the following exercise from Camacho & Neto's book $\textit{Geometric Theory of Foliations}$:
Let $\mathscr{F}$ be a $C^r$ $(r\geq1)$ $1$-dimensional foliation on $\mathbb{R}^2$ and $\pi:\mathbb{R}^2\longrightarrow\mathbb{R}^2/\mathscr{F}$ be the projection onto the leaf space of $\mathscr{F}$.
(a) Prove that all the leaves of $\mathscr{F}$ are closed, but that $\mathscr{F}$ has no compact leaves.
How can I show that there is no compact leaf? I believe the idea is to suppose that there is such a compact leaf $F$, conclude that it must be diffeomorphic to $S^{1}$ and then use Poincaré-Bendixson theorem on the vector field associated to the line field $P$ defined by the foliation in order to find a singularity. However, PB theorem asks for some regularity on the vector field and, if we just construct the vector field this way, we might not have any sort of regularity.
I also believe that the idea at this point is to go to the orientable double covering of $P$ and there apply Poincaré-Bendixson, but I really don't see how to do this. What am I missing?
Thanks in advance for any help!