Show that if X is a path-connected 1-dimensional CW complex with basepoint x0 a 0-cell, then every loop in X is homotopic to a loop consisting of a finite sequence of edges traversed monotonically. [See the proof of Lemma 1.15. This exercise gives an elementary proof that π1(S1) is cyclic generated by the standard loop winding once around the circle. The more difficult part of the calculation of π1(S1) is therefore the fact that no iterate of this loop is nullhomotopic.]