# Problem with the Pythagorean theorem [duplicate]

The Pythagorean theorem has already been proved and it is a basic fact of math. It always works, and there are proofs of it. But I have found a problem.

Say you want to get from point A to point B.

Here is a way to do it, where red is vertical movement and grey is horizontal movement.

Now say you split the path up like this. Note that it is the same length, as you can see from the color of the lines:

You can continue to do this... (note that the path still continues to stay the same length):

And if you continue forever, the path will become diagonal.

But now there's a problem. This is contradicting the Pythagorean theorem:

I know the Pythagorean theorem is true and proven, so what is wrong with this series of steps that I went through?

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## marked as duplicate by GEdgar, Thomas Andrews, Cameron Buie, Amzoti, 5pm Mar 8 '13 at 22:53

This is an abstract duplicate of this popular question, and indeed a direct duplicate of this question and this question. – Zev Chonoles Mar 8 '13 at 22:24
Length is a tricky notion. If you have two curves $y=f(x)$ and $y=g(x)$ that are visually indistinguishable from each other, ther area under one curve, from $x=a$ to $x=b$, is very close to the area under the other. But their lengths, as your work shows, can be quite different. – André Nicolas Mar 8 '13 at 22:26
@ZevChonoles Thanks, I didn't see those. You can close my question as a dup then. – Doorknob Mar 8 '13 at 22:27
In particular, see the accepted answer for the "duplicate" question 12906. – GEdgar Mar 8 '13 at 22:27

The problem here is that the limit of the lengths is not the length of the limit. One has assumed that the sequence of lengths $x+y,x+y,x+y,\ldots$ converges to the length of the hypotenuse in the fake proof.