The Messengers and their People's Sins

What do the sins of people have to do with the noble Messengers? Does this in any way defame their Prophet-hood in any way, or change anything in their message? Is any person from any of the different nationalities, other than the messengers –may peace be upon all of them, free of sin? Is it in anyway permissible to reject their message because of the sins of some of their followers?

This is not but the most evil of ways. It is like an ill person who is invited by a sincere doctor to take medicine that would cure him, but he says; ‘if you were a doctor there wouldn’t be any sick people.’

Is it binding upon the prophets to cure all sick people, to the point that there is not one sick person left on the face of the earth? Has anyone ever been this stubborn towards any of the prophets?!