Is the practice of "Tathwib" in the second Fajr Adhan Sunnah or Bid'ah?

The muezzin's saying "Prayer is better than sleep" in the Fajr adhan is a Sunnah and not an innovation. The proof for this is the hadith of Abu Mahdhurah in Musnad Ahmad and Sunan Abi Dawud. This saying was legislated specifically for the Fajr adhan because this time is a time of tranquility, and there is a fear of missing the Fajr prayer. So, this saying came as an encouragement, reminding of the importance and greatness of prayer, and that it is better than sleep, thus it is not permissible to abandon it for the sake of sleep.