The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?

Is $\pi=4?$

(non rigorous) If you repeat the process a million times it "seems" (visually) that the perimeter approaches in length to the circumference, but if you magnify the picture of a single "tooth" to full screen, you will notice a big difference from the orthogonal segments and the arc of the circumference. No matter how many times you repeat the process that difference will never fade.
ADDED: A visual example of what I meant is folding of a rope. If you imagine the rope not having thickness, you can fold it so many times that you can tend to a point (zero length?). If you unfold it, it will return to its original shape. In the example the perimeter will always be of total length = 4, but it only appears to blend with the circumference.
