Answer:
The easy way in short to prove the existence of Imam Mahdi (ajtf) to a Sunni is the following:
Premise 1: All Muslims agree that the Holy Quran is very explicit and clear that time will come that a universal just and Islamic state will be established when those whom been oppressed will lead the world. (Surah al-Anbi’a, Ayah 105).
Premise 2: It is narrated by Shi’a and Sunni narrators that:
a. The day of judgment will not come until a man from the offspring of the Prophet rises to spread peace and justice in the world. (Mosnad Ahmad 3:36 , H. 10920)
b. Whoever dies not knowing the Imam of his time has died the death of Ignorance (Kofr). (Muslim 3:239, H 1851)
c. Hadith al-Thaqalayn in which the Prophet (P) says: the Book of Allah and his Ahlul-Bayt (P) will never be separated until they join him at the Pool of Kouthar. (Mostadrak al-Hakim 3:10)
d. 12 Imams: The Prophet foretold that the Kholafa (successors after him shall be 12 Imams, that are all from Qoraysh). (Muslim 3:220, H. 1821). This hadith can only apply to the 12 Imams of Ahlul-Bayt (a.s). Also, there is no doubt that Imam Askari (a.s) was marytred and died.
Conclusion: if Imam Mahdi is not born yet, then: there will be a separtion between the Quran and Ahlul-bayt (a.s) (violating premise 2/c), the Imams won’t be 12 (violation of premise 2/d.) and all contemporary Muslims die the death of Jahiliyah because there is no Imam of their time for them to know him (violation of premise 2/b).
Answered by: Sheikh Mansour Leghaei