The Greatest and Holiest Prophet صَلَّى اللهُ تَعَالٰى عَلَيْهِ وَاٰلِهٖ وَسَلَّم has stated, ‘The fast of every day of Muharram is equivalent to the fasts of one month.’ (Al-Mu’jam-us-Sagheer, vol. 2, pp. 87, Hadees 1580)
صَلُّوۡا عَلَى الۡحَبِيۡب صَلَّى اللّٰهُ تَعَالٰى عَلٰى مُحَمَّد
Sayyiduna ‘Abdullah Ibn ‘Abbas رَضِیَ الـلّٰـهُ تَعَالٰی عَـنْهُمَا has stated, ‘I never saw the Noble Prophet صَلَّى اللهُ تَعَالٰى عَلَيْهِ وَاٰلِهٖ وَسَلَّم so enthusiastic to observe fast highlighting excellence to any day as much as he صَلَّى اللهُ تَعَالٰى عَلَيْهِ وَاٰلِهٖ وَسَلَّم would for the day of ‘Aashura and the month of Ramadan.’
(Sahih Bukhari, vol. 1, pp. 657, Hadees 2006)
The Holy Prophet صَلَّى اللهُ تَعَالٰى عَلَيْهِ وَاٰلِهٖ وَسَلَّم has stated, ‘Observe fast on the day of ‘Aashura, and oppose the Jews in this by observing an additional fast one day before or one day after it.’ (Musnad Imam Ahmad, vol. 1, pp. 518, Hadees 2154)
Whenever one is to observe the fast of ‘Aashura, it is better for him to fast on either the 9th or 11th of Muharram in addition.