For my differential topology course I need to prove that every vector bundle over $S^1$ is either trivial or equivelent to the Moebius bundle.
I found this nice, elementary proof: Line bundles of the circle, but it uses that every vector bundle over $\mathbb{R}$ is trivial. The proofs of this statement I found all use parallel transport, which we haven´t defined in the lecture and which we therefore shouldn´t use. Does anybody have a proof not using parallel transport or any other sophisticated tool like cohomology theory?