He can prove it by showing his birth certificate. He has to prove it, he is the president, not me.
Kerry may have done those things to smear the troops on film, but it was investigations that showed that his heroic claims were not true.
He doesn’t have to prove it. It’s required that you are natural born, but there are no legal requirements to show it. No federal court of appeals or the SCOTUS would rule that his COLB is insufficient to satisfy natural born citizen requirements. None.