 The proof can be given by induction after "n", with the statement that "f" is cellular on the skeleton "X""n".For the base case n=0, notice that every path-component of "Y" must contain a 0-cell.The image under "f" of a 0-cell of "X" can thus be connected to a 0-cell of "Y" by a path, but this gives a homotopy from "f" to a map which is cellular on the 0-skeleton of X.