A consequence of Hurwitz's theorem is that the right-hand side of the above inequality cannot be improved by an arbitrary positive constant s. More precisely, for s < l / x / 5 there exist real numbers a E I for which the inequality Ilqall ~<cq -1 has at most a finite number of solutions. These a are the badly approximable numbers, and we will denote by B a d the set of all such numbers; that is,