Prove that for every real vector bundle over $S^1$ there exist open connected subsets $U_1,U_2 \subset S^1$ with $U_1 \cup U_2=S^1$ such that $E$ is trivial over $U_1$ and over $U_2$.
I need an elementary proof of this fact. I know that v.b. over contractible spaces are trivial, but we do not have seen this result during lecture about smooth manifolds, and we are supposed to prove this fact.
Other than basic definitions, we covered sections, frames, whitney sums, inner products, and the facts that every bundles is a subbundle of a trivial bundle over the same base space. With so few instruments I do not know where to start to prove this fact.