Define the Mobius strip $E$ to be the quotient $\mathbb{R}^{2}/\sim$ where
$\left(x_{1},y_{1}\right)\sim\left(x_{2},y_{2}\right)$ if and only if $$\left(x_{2},y_{2}\right)=\left(x_{1}+2\pi k,\left(-1\right)^{k}y_{1}\right)$$ for some integer $k$. It is a smooth vector bundle over $\mathbb{S}^{1}$ whose fiber at every point $e^{i\theta}\in \mathbb{S}^{1}$ is $E_{\exp\left(i\theta\right)}=\left\{ \left[\left(\theta,y\right)\right]:y\in\mathbb{R}\right\}$ with the obvious vector space structure (here, $\left[\left(\theta,y\right)\right]$ is a typical element of $E$) . I want to show that this smooth vector bundle is nontrivial. This is equivalent to showing that every smooth global section $s:\mathbb{S}^{1}\rightarrow E$ must vanish somewhere. This is how I proceeded:
Suppose that there exists a smooth global nowhere vanishing section $s:\mathbb{S}^{1}\rightarrow E$. We seek to find a contradiction by showing that $s$ must vanish somewhere. Define a smooth map $F:\left[0,2\pi\right)\rightarrow\mathbb{R}$ as follows. Given a real number $x\in\left[0,2\pi\right)$ , define $F\left(x\right)$ to be the real number such that $s\left(\exp\left(ix\right)\right)=\left[\left(x,F\left(x\right)\right)\right]$. It remains to show that there exists a real number $x\in\left[0,2\pi\right)$ such that $F\left(x\right)=0$. This will complete the proof.
I am stuck at this point. I think that I might have to use the intermediate value theorem to show that $F\left(x\right)=0$, so it suffices to show that $F$ changes sign somewhere. I am unsure how to show this however.
