The manner in which the vertices are enumerated is important. I'm guessing you started with an arbitrary vertex and called it , then repeated the following step--for every vertex that's not already in the list, and which shares an edge with a vertex that is in the list, add it to the end of the list--until every vertex is in the list, which is guaranteed to happen eventually (possibly after infinitely many steps, but countably infinitely many, so that the number of vertices already in the list at any given step is finite) by connectedness of the graph.

Clearly every vertex was added at some step, and was added because it shared an edge with a vertex already in the list, i.e. a vertex with lower index.