The proofs I have seen that a vector field on the 2-sphere must have a zero rely on the general theorem that the index of any vector field on a manifold equals the manifold's Euler characteristic. How about this for a proof that does not appeal to this general theorem? The tangent circle bundle of the 2 sphere is homeomorphic to real projective 3 space. A non-zero vector field would be a map from [tex] S^2 -> RP^3 [/tex] which has a left inverse, p, where p is just the bundle projection map. Otherwise put, p o v = identity on [tex] S^2 [/tex] But the second real homology of projective 3 space is zero so p o v must equal zero on the fundamental cycle of the 2 sphere. This contradicts the equation p o v = identity.