I'm trying to understand why there are no non-vanishing sections in $\mathbb{R}P^n$.
We have the canonical line bundle over $\mathbb{R}P^n$ given by $E:=\{(l, v) \in \mathbb{R}P^n \times \mathbb{R}^{n+1}| v \in l\}$ with the map $\pi$ being just the projection on to the first coordinate and a section is just picking out vectors in $\pi^{-1}(l)$ for each $l\in \mathbb{R}P^n.$ $\pi^{-1}(l)$ is just $\mathbb{R}$, right? I then don't see what's wrong with having a non-vanishing section in $\mathbb{R}P^n$. I'd appreciate if someone could point out any mistakes I'm making or where to go from here.