That proof is just plain incorrect, though. It will break down when trying to prove this statement:

>Perimeter of the cut square (4) matches the circumference of the circle (pi)

Calculus will show that the area of the fractal approaches the area of the circle. But it will not show that the perimeter of the fractal approaches the circumference of the circle. It remains 4 at every step in the iteration, so the limit is still 4.