The best approach in proving the Imamate of Imam Ali (as) for followers of the Sunni school of thought is through means of what has been mentioned in the Quran and narrations which have been mentioned in their own sources. Obviously, this can only be accomplished when an individual is willing and eager to seek the truth. Meaning, one engaged in such a quest should be prepared to turn away from that which is false or irrational.
The Imamate of Imam Ali (as) can be proven through means of many verses found within Holy Quran and narrations from the Prophet (pbuh) surrounding their revelation.
Consider the following verses:
1. The Verse of Tabligh (propagation): “O Apostle! Communicate that which has been sent to you from your Lord, and if you do not, you will not have communicated His message, and Allah shall protect you from the people. Indeed Allah does not guide the faithless lot.”[i] According to authentic Sunni narrations, this verse was revealed on the day of Ghadir Khum. In this historical day, the Prophet announced the wWilayah of Imam Ali (as), thus completing his prophet-hood and making the Imam his successor.
2. The Verse Wilayah: “Your guardian is only Allah, His Apostle, and the faithful who maintain the prayer and give the zakat while bowing down.”[ii] This verse places the Wilayah of Imam Ali (as)[iii] alongside that of Allah and the Holy Prophet (pbuh).
3. The Verse of Ulil-amr: “O you who have faith! Obey Allah and obey the Apostle and those vested with authority among you. And if you dispute concerning anything, refer it to Allah and the Apostle, if you have faith in Allah and the Last Day. That is better and more favorable in outcome.”[iv]
This verse states that following the leadership of the Ulil-amr is obligatory. From this we can also deduce that such leadership requires the Imam to be infallible, a quality possessed by none other than Ali (as). In reference to this verse, there are numerous narrations in Sunni sources which introduce Imam Ali (as) as the successor of the Prophet (as) and Imam of the believers.
4. The Verse of Sadiqin: “O you who have faith! Be wary of Allah and be with the Truthful.”[v] It has been mentioned in narrations that the word “Truthful” mentioned in this verse is in reference to Imam Ali (as) and Prophet’s household.
The best approach in proving the Imamate of Imam Ali (as) for followers of the Sunni school of thought is through means of what has been mentioned in the Quran and narrations which have been mentioned in their own sources. In order to do this, one should thoroughly understand that which is presented in verses of the Quran. Meaning, the content of the verse, the context in which it is introduced, instances it refers to, etc. should all be considered in detail.
In the Holy Quran there are many verses which make reference to and establish the Imamate of Ali (as) and the other Imams. Concerning what has been mentioned in these verses, there are innumerable narrations from the Holy Prophet (pbuh) which expand further on the circumstances surrounding the revelation of these verses and their details. These narrations are considered authentic by both the Shia and Sunni schools of thought.
The follow are a few of those verses and the appropriate narrations concerning them:
1. The Verse of Tabligh (propagation): “O Apostle! Communicate that which has been sent to you from your Lord, and if you do not, you will not have communicated His message, and Allah shall protect you from the people. Indeed Allah does not guide the faithless lot.”
In this verse, God ordered the Prophet (pbuh) to communicate his message to the people with heavy emphasis. According to narrations, this verse was revealed in a place by the name of “Ghadir Khum”. In this place, the Prophet (pbuh) publically announced Ali (as) as his successor and the first Imam. The event of Ghadir took place in the last year of the Holy Prophet’s life during the time of Hajjatul Vida (the Prophet’s final Hajj) where he stated: “من کنت مولاه فهذا علی مولاه...”, meaning, “For whomever I am his master, Ali is master…”.
This event has been narrated by 110 companions of the Prophet (pbuh), 84 tabiyin (those who have been informed through intermediaries), and 36 scholars and writers. Allameh Amini, in his book Sharife al-Ghadir, has mentioned all of these narrations with cogent evidence from Islamic sources (authenticated by both the Shia and Sunni schools of thought).
2. The Verse Wilayah: “Your guardian is only Allah, His Apostle, and the faithful who maintain the prayer and give the zakat while bowing down.”
Many interpreters and narrators maintain that this verse was revealed in regards to the elevated status of Imam Ali (as)
Suiti (a Sunni scholar), narrates from the book Dar al-Mansour (a book containing commentary of the Quran), quoting Ibn Abbas who narrates: “Ali (as) was in the ruku of his prayer when a beggar came along asking for help. Ali (as) then proceeded (as) to give the ring on his hand to the beggar as charity. The Prophet (pbuh) asked the beggar, “Who gave you this ring as charity?” He pointed towards Ali (as) and said, “That man who is in ruku did.” At that moment, the verse mentioned above was revealed.
In his commentary, Fakhr Radhi narrates from Abdullah bin Salaam that: “When this verse was revealed, I said to the Prophet of God (pbuh) that I saw with my own eyes that Ali (as), while in ruku, gave his ring as charity to a needy person. For this reason we accept his wilayah!” In another narration, he also quotes Abuzar who narrates the same occurrences surrounding the revelation of this verse.”
In his tafsir, Tabari cites many narrations associated with this verse and the circumstances surrounding its revelation. A majority of those narrations affirm that the verse was revealed in regards to Imam Ali (as).
In his book al-Ghadir, the late Allamah Amini has quoted this verse and its association with Imam Ali (as), citing narrations found in approximately twenty Sunni books of narrations (along with their specific chains of narration and sources).
It is in this verse that the wilayah of Imam Ali (as) is mentioned alongside that of Allah and the Holy Prophet (pbuh).
3. The Verse of Ulil-amr: “O you who have faith! Obey Allah and obey the Apostle and those vested with authority among you. And if you dispute concerning anything, refer it to Allah and the Apostle, if you have faith in Allah and the Last Day. That is better and more favorable in outcome.”
Scholars have stated that this verse has been revealed in reference to Imam Ali (as).
For example Hakem Haskani Hanafi Neishabouri (a renowned Sunni interpreter of the Quran) has quoted five narrations for this verse, and of which associate the title of “Ulil-amr” with Imam Ali (as). 
The commentary of al-Bahr al-Muhit, written by Abuhayan Andulusi Maghribi, quotes Muqatil, Meimoun, and Kalbi (all interpreters of the Quran) who agree that this verse is in reference to the Imams (as) from the household and lineage of the Prophet (pbuh).
In his book Eteghad, Abu Bakr bin Mumin Shirazi (a Sunni scholar), quotes Ibn Abbas who affirms that the above verse was revealed in regards to Imam Ali (as).
In brief, this verse informs mankind of its obligation in obeying God, the Prophet (pbuh), and the Ulil-amr. Since this order has been made wajib (obligatory) to the highest degree, it can be inferred that the Ulil-amr, like God and his Prophet (pbuh), should also possess infallibility. Considering what has been mentioned in narrations, this could only be in reference to the infallible twelver Shia Imams (as).
What has been mentioned here, are only a few of the authentic narrations found in Sunni books and three verses found in Shia books regarding the Wilayah and Imamate of Ali (as). There is an abundance of other topics and points to be discovered in many Shia and Sunni books alike. For further information, one should refer to these reliable sources.
In addition to the three verses which have been mentioned, there are others which directly establish the Wilayah and Imamate of Ali (as) and the other Imams (as). The following are a couple of those verses: The verse of Sadiqin which reads “O you who have faith! Be wary of Allah and be with the Truthful.” and the verse of Gurba which reads “That is of which Allah gives the good news to His servants, (to) those who believe and do good deeds. Say: I do not ask of you any reward for it but love for my near relatives; and whoever earns good, We give him more of good therein; surely Allah is Forgiving, Grateful.” These verses, in addition to narrations from the Holy Prophet (pbuh), can be found in both Shia and Sunni sources.
Besides the mentioned verses, there are others which highlight the virtues and qualities of Imam Ali (as) and depict his superiority in comparison to the other companions and supporters of the Holy Prophet (pbuh). Logically speaking, it is absurd to select that which is inferior as opposed to that which is superior. Therefore, it is only was rational for Ali (as) to succeed the Holy Prophet (pbuh) and take his rightful role as Imam. For further information on this matter, you may refer to volume nine (Imamate and Leadership in the Holy Quran, page 177 and on) of Ayatollah Makarem Shirazi’s commentary of the Holy Quran.
To see why the name of Imams are not mentioned explicitly in Quran please refer to the subject: The name of Imams (AS) in Quran
For more information please refer to the subject: The reasons to believe Imamate and Imams
Maedeh 67 for more information refer to the subject: Sunnis and the verse "Tabligh"
Makarem Shirazi, the message of Quran, vol. 9, Imamate in Quran, from page 182 on.
Abu saeed Khadri, Zein Al din Argham, Jaber bin Abdullah Ansari, Ibn Abbas, Bara bin Aazeb, Hazifeh, Abu harireh, Ibn Masoud and Amer bin Laili.
Aldorr Almanthour, vol. 2 page 293
Asbab Al Nozool page 148
Tafsir Keshaf vol. 1, page 649
Tafsir of Fakhr Razi
Tabari Tafsir, vol. 6, page 186
Alghadir, vol. 2, page 52, 53
There are tens of hadiths in sources of house hold of the prophet below this verse: The mentioned verse has been descended for Ali (AS) and other Imams even in some verses the names of twelve Imams have been mentioned one by one. Borhan Tafsir vol. 1, page 381 – 387.
Shavahed al Tanzil vol. 1, page from 148 to 151.
Bahr Al Mohit. Vol. 3, page 425
Ihqaqul-Haqq, vol. 3, page 425
Tobeh, 119, we have in hadiths that: what is meant by "true in words and deeds" are Ali (AS) and house hold of the prophets. (Yanabe al Mavadat page 115 and Shavahed al Tanzil vol. 1 page 626)
Shora 33, for more information, refer to subject: The house hold of the prophet (PBUH) and the verse Mawaddah.