Doubly eulerian trails on rectangular grids

Let K be the 1-skeleton of the regular tessellation of Euclidean n-space by n-cubes, n ≥ 2. We show that K admits a doubly Eulerian trail (simply Eulerian trail), that is, a doubly infinite path π = … e_?1e₀e₁ … where, out of each pair {e, e^?1} of oppositely directed edges, both (exactly one) appear(s) exactly once in π, and where no e_i+1 = e_i^?1 (there are no U-turns).