I have a problem showing that any vector bundle $\displaystyle E \to [0,1]$ is trivial.

The hint given with the exercise is considering the biggest number $\displaystyle t \in [0,1]$ so that $\displaystyle \pi^{-1}([0,t]) \to [0,t]$ is trivial, but i don't get how to do that, in fact I'm not even sure what that last statement means.

So far I think I've shown that for some $\displaystyle t \in (0,1)$, we have a homeomorphism $\displaystyle \pi^{-1}([0,t]) \to [0,t]$, since there must exist a homeomorphism to an open neighbourhood of 0, and we can restrict that. Also, $\displaystyle [0,t]$ is homeomorphic to $\displaystyle [0,1]$. But where do I go from there?