@dhy The paper is
arxiv.org/abs/math/0303382 and it is Lemma 4. I can fix this, so it is not that terrible, but I don't see the proof now as correct. The fix I have is to base change to the algebraic closure, where you easily can find a section not vanishing (since a vector space over an infinite field is not a finite union of proper subspaces). Then, we take this section s and the product of all its Galois conjugates and then descend