Let G be an undirected graph. Suppose you start at a vertex v and walk along the edges of G obeying the following two rules.
(I) You may not use the same edge twice travelling in the same direction (so you may use each edge at most twice total, once in each direction).
(II) You must continue walking until no further moves are permitted (due to the first rule).
Prove that once your walk stops, you are at v.
I considered replacing each edge with 2 directed edges in each direction, but I still draw a blank.
Hint: Having replaced each undirected edge with a pair of directed edges, each vertex now has in-degree equal to its out-degree. Imagine deleting each directed edge as you use it. Your walk comes to an end when you reach a vertex whose out-degree is $0$, which means its in-degree was greater (by at least one) than its out-degree. How can this happen?