I'm reading Spivak's Comprehensive introduction to differential geometry and i came across the proof that the tangent bundle of the Möbius band (as he defines it at an early stage i presume) is not homeomorphic to $M\times\mathbb{R}^2$. Here is the complete proof. 
I got the first part of the proof. What i don't get is the conclusion where he says that it is clearly impossible to map etc... Here my attempt: Let's suppose that there exist an homeomorphism such that takes fibre into fibre in a linear fashion $$\phi:T(M,i)\to M\times\mathbb{R}^2$$ For each $\theta\in[0,2\pi]$ consider the subspace $<f_{*}(v_{(\theta,0)})>$ where $v=(0,1)$. This subspace is mapped by $\phi$ onto a subspace of $\mathbb{R}^2$. Then i have a collection of subspace of $\mathbb{R}^2$ parametrized by $\mathbb{S}^1$ and i wish to pick for each point in $\mathbb{S}^1$ a non zero vector on $\phi(<f_{*}(v_{(\theta,0)})>)$ in a continuous fashion. But how? Is it even possible? This is equivalent to lift to $\mathbb{S}^1$ a curve in $\mathbb{P}^2$.