How prophet (SA) himself started a new month - هَاتُوا بُرْهَانَكُمْ‎

Prophet (SA) instructed sahabas to do Salath (Namaz), Fasting, Hajj etc and showed them how to perform by doing himself.

All will agree that.

Public belief is that islamic tradition of starting month is, by sighting waxing crescent at western horizon after sunset. If so, that tradition must have started from the practice of prophet (SA) himself.

As prophet (SA) himself did what he instructed, we would like to see evidence of beginning of month 'by sighting waxing crescent at western horizon after sunset' by prophet himself. 

Had ever prophet gone himself to search so called 'hilal' for beginning month . we are not asking evidence of what somebody else had done. Very specific how prophet himself demonstrated and did.

As prophet (SA) himself performed Salath (Namaz), Fasting, Hajj etc which he taught, we would like to know what he did himself about beginning of month.

Also quran warns to all, including prophet, as: 61:2,3
يَا أَيُّهَا الَّذِينَ آمَنُوا لِمَ تَقُولُونَ مَا لَا تَفْعَلُونَ ﴿٢﴾ كَبُرَ مَقْتًا عِندَ اللَّـهِ أَن تَقُولُوا مَا لَا تَفْعَلُونَ ﴿٣

  ﴾O you who believe! Why do you say that which you do not do? . Most hateful it is with Allah that you say that which you do not do. (Al-Quran 61: 2 to 3)

This is the strict order of Al-Mighty Allah and we believe that our beloved Messenger of Allah (saw) strongly adhered to it.

During His (saw) prophet hood age (23 Hijri years) he met 10 obligatory fasting Ramadhans and two Haj performances. That is 10 Ramadhan beginning and 10 Shawwal beginning to celebrate 10 Eidul Fitr festivals, in which fasting is Haram as per his (saw) own instructions. For two Haj it is obligatory to stand at Arafa on 9th Dul Hijjah. Therefore, Our Messenger of Allah had exactly 22 chances to see the first Hilal by naked eye in order to perform the above noted obligations.

Either He (saw) would have seen the first Hilal personally or ordered his companions to search out and inform it to him. Is there any evidence either from Quran or Ahadhees?

This is easy way to solve our problem in starting the Hijri month especially Ramadhan and Haj, which has obligatory blessings.

Kindly provide evidence of the "practise of starting month after SIGHTING CRESCENT at WEST after sunset" by prophet (SA) himself.