If he cannot be impeached because he is not president then how can he go on being president?
He’s president because he was sworn in as president. You could swear in a bowl of petunias, and then it would be president.
A better president.
But the political implications of having run and then certified an ineligible candidate for the Presidency would be huge and unprecedented. Obama would have to step down (or risk civil war), and the Democrats and all their minions would be permanently stained. 30% of the country would continue to vote for them, but that 30% are poltroons anyway.
He WOULD NOT be President. He just may stay on in the White House, until EVICTED in disgrace.