This is an extract from the book "Al Jaami'u li Ahkaam is Siyaam" (A complete guide to the rules of fasting) by Sheikh Abu Iyaas Mahmood bin Abdul-Lateef bin Mahmood ('Uwaydhah). Please note this is a draft translation from Arabic. If any confusion arises from the translation then please refer to the original Arabic book.
In relation to fasting the first nine days of Dhul Hijjah then Al-'Amal As-Saalih (the righteous action) of which fasting is included is better in this nine days than Jihaad in the way of Allah except that a Mujaahid expends his body and wealth in it.
Allah سبحانه وتعالى has sworn by the ten nights of Dhul Hijjah in the first four ayaat of Surah Al-Fajr:
وَاللَّيْلِ إِذَا يَسْرِ
"By the Dawn,
And ten nights,
And the Even and the Odd,
And the night when it departs"
Imaam Ahmad (14565), An-Nasaa'i, Al-Bazzaar, At-Tabari, Ibn ul-Mundhir, Al-Bayhaqi and Al-Haakim who verified it as Saheeh narrated from Jaabir (ra) that the Prophet صلى الله عليه وسلم said:
"Verily the ten are the ten of Al-Ad'haa (Dhul Hijjah), the Witr (odd) is the day of 'Arafah and the Shaf'i (even) is the day of slaughtering".
The following are a collection of Hadeeth that encourage the increase of good actions in it including As-Sawm:
a) Ibn Abbaas (ra) said that the Messenger of Allah صلى الله عليه وسلم said:
"There are no days in which the righteous acts are more beloved to Allah than in these days, meaning the ten (of Dhul Hijjah). They asked: O Messenger of Allah صلى الله عليه وسلم, not even Jihaad in His سبحانه وتعالى way? He صلى الله عليه وسلم replied: Not even Jihaad in the way of Allah except for the man that goes out with his life and his money and does not return with either of them".
Ibn Maajah (1727), Abu Daawud, Ahmad, Ad-Daarami and Al-Bayhaqi.
At-Tirmidhi said that this hadeeth is: Hasan Ghareeb Saheeh.
At-Tabaraani (1777) in al-Mu'jam Al-Awsat narrated it from Ibn Mas'ood (ra) and (4398) from Abu Qataadah (ra).
b) Hunaidah bin Khaalid narrated from his wife from some of the wives of the Prophet صلى الله عليه وسلم:
"That the Messenger of Allah use to fast nine from Dhul Hijjah, the day of 'Aashoorah and three days from every month: The first Monday and the following two Thursdays".
An-Nasaa'i (2417). Imaam Ahmad (26991) narrated the following Lafzh: That Hafsah (ra) said:
"There are four actions that the Prophet صلى الله عليه وسلم did not leave: the fast of 'Aashoorah, the ten (days of Dhul Hijjah), three days from every month and two Rak'ah before al-Ghadaa".
Abu Daawud (2437) narrated: "The Messenger of Allah صلى الله عليه وسلم would fast the nine days of Dhul Hijjah, the day of 'Aashoorah, three days from every month: the first Monday of the month and Thursday".
In relation to what was narrated from 'Aa'isha (ra) that: "I did not see the Messenger of Allah صلى الله عليه وسلم fasting in the ten (days of Dhul Hijjah) at all" recorded by Muslim (2789), Abu Daawud, An-Nasaa'i, At-Tirmidhi, Ibn Maajah and Ahmad.
Then this reflects the extent of the knowledge of 'Aa'isha (ra) in this issue. Her statement is negating whereas the other Ahaadeeth are affirming and the affirmation (Ithbaat) is always stronger than the negation (Nafy). The one who saw and knew about a thing has a proof over the one who did not see or know about it and therefore this hadeeth cannot abrogate the Ahaadeeth which affirm the fasting in these ten days.