Continuing in this fashion, an infinite simple path can be constructed using mathematical induction and a weak version of the axiom of dependent choice. At each step, the induction hypothesis states that there are infinitely many nodes reachable by a simple path from a particular node "v"i that does not go through one of a finite set of vertices. The induction argument is that one of the vertices adjacent to "v"i satisfies the induction hypothesis, even when "v"i is added to the finite set.