TUYMAADA 2022 J6

The city of Neverreturn has $N$ bus stops numbered $1, 2, \cdots , N.$ Each bus route is one-way and has only two stops, the beginning and the end. The route network is such that departing from any stop one cannot return to it using city buses. When the mayor notices a route going from a stop with a greater number to a stop with a lesser number, he orders to exchange the number plates of its beginning and its end. Can the plate changing go on infinitely? (K. Ivanov )