He didn't prove the wiggly-sided paralellogram -> rectangle transition when the angle of each slice goes to 0. He also did not prove that he is allowed to take

He didn't prove the wiggly-sided paralellogram -> rectangle transition when the angle of each slice goes to 0. He also did not prove that he is allowed to take the angle to 0. I can grant him that any angle > 0 results in a less and less squigglier paralellogram with more and more vertical sides with an obvious proof, but that doesn't prove either that the limit is a rectangle, or that reasoning at the limit is transitive to reasoning about the circle. I don't doubt the result or the method - they both seem intuitively correct, they just aren't proven.