Give me your argument on this square (what seems like) paradox?

This is something i came up with while bored on an airplane. It will be difficult to explain so read carefully. How long is the diagonal of a square? Everyone knows it is x sqrt2 where x is the length of one side. Suppose now that we want to cut smaller squares out of this square to create a staircase. The 1st step is cutting a square 1/4th the area of the original square away from one of the corners. We now have two "steps". We now cut squares 1/8th the area of the original square away from the corners of our steps. Two steps, so two corners, so we cut away 2 of these squares. If we do this, we see that we now have a 4 step staircase. If we repeat this process ad infinitum we end up with infinitesimally small steps that are indistinguishable from the diagonal. Yet, if you calculate the length of our staircase, you will discover that it will always be equal to 2x (I won't do the math here, it's pretty obvious). Can you see the problem already? Let's take it one step further: You could make the argument that underneath the staircase there is more area, but actually if you take the limit to infinity, you will see that the area converges to 1/2 x^2 which is the same as if there was just a diagonal and not a staircase. So you have gotten this far; here is the question: how can the diagonal be two different lengths even though it encloses the same area?