Proof That Pi = 2: Intuitively Wrong?

[adriank]

Of course that's not true, and I never said that.

 

[Dickfore]

Of course that's not true, and I never said that.

I edited my post.

 

[adriank]

Right, and it's precisely because $\gamma_n'(t)$ doesn't converge that the lengths don't converge to the length of the limit. But every length involved exists.

This is a classic example where some behaviour that is constant during some limiting process is not preserved once you take the actual limit. (In other words, the "length of a path function" is not "continuous".)

 

[Dickfore]

I was simply arguing that the lengths do not converge at all, not just to that particular value.

 

[adriank]

The derivatives don't converge, but the lengths certainly do, since they're the constant $\sqrt2$! They don't converge to 1, that's all.