How to prove that any vector bundle on a Euclidean space is a trivial bundle? It is enough to prove it for the case of dimension $1$ and I hope it will be a nice exercise for me to generalize to the higher dimensional case.
What is the underlying topological reason that all bundles are trivial? Certainly simply connectedness will not do, as a counterexample is given by the Hairy ball theorem(of which I know only the statement; not the proof). Is it enough that the space is contractible?