Every point on the limit of the fractal is within epsilon of some point on x^2 + y^2 = 1, for every epsilon > 0. So, if some point on the limit of the fractal

Every point on the limit of the fractal is within epsilon of some point on x^2 + y^2 = 1, for every epsilon > 0. So, if some point on the limit of the fractal were not on the circle, it would be some finite distance d from the circle. By choosing epsilon = d/2 you have a contradiction. The circumference of the limit doesn't equal the limit of the circumferences.