I want to make a project at differential geometry about the Hairy Ball theorem and its applications. I was thinking of including a proof of the theorem in the project. Using the Poincare-Hopf Theorem seems easy enough, but I was thinking that this proves the desired result using a stronger theorem (just like proving Liouville's Theorem in complex analysis using Picard's theorem).
Is there a simple proof of the fact that there is no continuous non-zero vector field on the even dimensional sphere? It is good enough if the proof works only for $S^2$, because that is the case I will be focusing on in the applications.