Math Help - Trivial vector bundle problem

1. Trivial vector bundle problem

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

The hint given with the exercise is considering the biggest number $t \in [0,1]$ so that $\pi^{-1}([0,t]) \to [0,t]$ 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 $t \in (0,1)$, we have a homeomorphism $\pi^{-1}([0,t]) \to [0,t]$, since there must exist a homeomorphism to an open neighbourhood of 0, and we can restrict that. Also, $[0,t]$ is homeomorphic to $[0,1]$. But where do I go from there?

2. I think it's going to go something like

1) Show such a t exists.

2) Show t=1 (assume <1 and get a contradiction).

I suspect that connectedness and/or compactness will come into play, but I can't see the answer immediately.