Well history is written by winner so "the people" will accept what they're told, they have no power, and no reason to not to accept it.

As for "how you prove it", they lie. Jon says Ned told him before he left for the wall, and Jon didn't mention it to anyone after that for very good reason (it's a death sentence), and Danny lies, saying her brother told her, and Varis lies about it, saying it information he's been sitting on for years.

So three people with corroborating stories, who have been party to the same secret.

Well history is written by the winner so "the people" will accept what they're told. They have no power, and no reason to not to accept it.

As for "how to prove it", they lie. Jon says Ned told him before he left for the wall, and Jon didn't mention it to anyone after that for very good reason (it's a death sentence), and Dany lies, saying her brother told her, and Varys lies about it, saying it information he's been sitting on for years.

So they'd have three people with corroborating stories, who have been party to the same secret.