This little geometric riddle is something that I have thought about way too much, and it's driving me crazy. I know this is incorrect but I can't figure out why:

The diagonal of a square is twice the length of its sides.

Crazy, right? We all know it's not true. The Pythagorean Theorem states that for a triangle with sides

*a*,

*b*, and hypotenuse

*c*,

*a*^{2} + b^{2} = c^{2}

which means that for triangles formed by a diagonally bisected square with sides of length

*n*,

*n*^{2} + n^{2} = c^{2}

where c is the length of the diagonal. Simple algebra gives us

*c = √2n*

That's what we all learned in geometry, right? The length of the diagonal of a square is √2 times the length of its sides. Physical reality backs up this fact. Grab a ruler and see for yourself. Now then, why does the following "proof" make sense?

Assuming a square with sides of length

*a*, the distance between opposite corners of the square traveling along the border of the square is

*2a*.

By dividing both sides into two equal segments and alternating directions, we create a new path of the same distance

*2a*.

*2(a/2) + 2(a/2) = a + a = 2a*

By continually dividing each side into

*n* equal segments of length

*a/n* and rearranging them in this manner, we create a path that is arbitrarily close to the diagonal without changing the total length of the path:

*n(a/n) + n(a/n) = a + a = 2a*

As the number of segments

*n* approaches infinity, the total length remains unchanged.

*lim*_{(n → ∞)} 2an/n = lim_{(n → ∞)} 2a = 2a

Thus the length of the diagonal is

*2a* or twice the length of a side.

Please, someone tell me why I'm wrong.