The Ahl Al-Bayt taught us Muslims to fast on Ashoora, Ahl Al-Sunnah are the true followers of the Ahl Al-Bayt:

al-Bahrani in Hadaiq 13/370 said:

فاما ما يدل على استحباب صومه فمنها ما رواه في التهذيب عن ابى همام عن أبى الحسن (عليه السلام) قال: ” صام رسول الله صلى الله عليه وآله يوم عاشوراء “

As what is a proof for fasting (in this day) being mustahab is what narrated in Tahzib from Abu Humam from Abul Hasan (alaihi salam) which said: Prophet – sallalahu alaihi wa alihi fasted on the day of Ashoora.

وما رواه عن عبد الله بن ميمون القداح عن جعفر عن ابيه (عليهما السلام) قال: ” صيام يوم عاشوراء كفارة سنة “.

And what was narrated from Abdullah ibn Maimun al-Qaddah from Jafar from his father (alaihuma salam) which said: Fast of day of Ashoora an atonement of the year.

وما رواه الشيخ عن مسعدة بن صدقه عن ابى عبد الله عن ابيه (عليهما السلام) ” ان عليا (صلوات الله وسلامه عليه وآله) قال: صوموا العاشوراء التاسع والعاشر فانه يكفر ذنوب سنة “

And what was reported by shaykh from Musaddat ibn Sadakah from Abu Abdullah from his father (alaihuma salam): Ali (salawatullah wa salamuhu alaihi wa alihi) said: Fast on Ashoora, 9 and 10, that is atonement of sins of the year.

Regarding second hadith al-Khui said it is saheeh, and regarding 3-d one, al-Khui said it is muwathaq. See Kitab as-Sawm 2/305