I'm stuck with something I've been trying to prove:
A tautological 1-dim vector bundle $\pi:E\rightarrow\mathbb{R}P^n$ is given, where $E\subset \mathbb{R}P^n \times\mathbb{R}^{n+1}$ is a subset of pairs $([x_1,...x_{n+1}],v)$, such that $v\in[x_1,...x_{n+1}].$ Note: $\mathbb{R}P^n$ is a $n$-dimensional projective space, elements of which we denote as $[x_1,...,x_n].$
I want to show that there are no global non-vanishing section $s:\mathbb{R} P^n\rightarrow E$, such that $s(m)\neq 0$, for all $m\in\mathbb{R}P^n$.
My thinking (I'm a total beginner, please be kind):
$E\subset \mathbb{R}P^n \times\mathbb{R}^{n+1} $ probably means that $E$ is a (vector) subspace of $\mathbb{R}P^n \times\mathbb{R}^{n+1}$, meaning that it has to contain a $0$ (vector). We know that the global section is defined such that $\pi \circ s=I_{d|\mathbb{R} P^n}$, in order for this composition to work $s(m)$ needs to be $s(m)=0$ for some $m\in\mathbb{R} P^n$, otherwise the codomain of $s:\mathbb{R} P^n\rightarrow E$ and the domain of $\pi$ above don't match and we can't have $\pi \circ s$:
On my second try, I tried to use the fact that sections are smooth maps and maybe deduct something useful, but to no avail.
I would really appreciate some help.
EDIT: as someone in the comments suggested my terminology regarding the vector (sub)space is wrong, the correct term would be (sub)bundle.
Take your section $s$. It corresponds to a unique map $\widetilde{s}:\Bbb S^n \to \Bbb R^{n+1}$, such that $\widetilde{s}(-x)=\widetilde{s}(x)$ and $\widetilde{s}(x)\in \Bbb R x$ for all $x$. This means you can write $\widetilde{s}(x)=\mu(x)x$, where $\mu:\Bbb S^n\to \Bbb R$ corresponds uniquely to $\widetilde{s}$ (and hence $s$) and satisfies $\mu(-x)=-\mu(x)$ for all $x$. By Borsuk-Ulam there is $x_0\in\Bbb S^n$ with $\mu(x_0)=\mu(-x_0)$. So $\mu(x_0)=0$ and so $s(\Bbb R x_0)=0$ as wanted.