A recursive construction for universal cycles of 2-subspaces

We prove that universal cycles of 2-dimensional subspaces of vector spaces over any finite field FF exist, i.e., if VV is a finite-dimensional vector space over FF, there is a cycle of vectors v1,v2,…,vnv1,v2,…,vn such that each 2-dimensional subspace of VV occurs exactly once as the span of consecutive vectors.