This is the whole argument in a nutshell.

Because there is no way to see all possible things in reality at once (except to see the single fundamental principle of reality), no one can say such and such a thing is real. It is overwhelmingly more likely that your limited perception of eternity is in fact not what eternity is 'made of'. It is because infinity, eternity, is unequivocally not constructed out of finite pieces, that the most logical and most in line with direct experience conclusion is "We are not perceiving reality as it is, but only as we perceive it."

If you wager that reality, existence itself, is not eternal/infinite, you've got an endlessly high mountain to get to the top of before you can say its finite.