Proof that sqrt 2 = 2

Consider a square with opposing corners A and B:

 

There are different possible paths from A to B. One can travel first along the x-axis and then along the y-axis, making the total travel distance 2 times the side of the square:

One can also travel along the diagonal

 

and by using the Pythagora theorem, that distance is shorter by a factor

Now, consider a third path, go half along x, half along y, half along x and then half along y:

Again clearly a a total distance 2 since you just folded the blue path on two places. You can divide this up in more slices, but the total distance will remain constant 2, e.g:

However, in the limit, the path taken will be that of the diagonal, which is

shorter. But we know the path is 2 long since all such paths are 2 long, and hence

since

 

 

where sn is the distance with n divisions of the path. What is wrong?