[chemotaxis]

I think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".

There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.

[JKCalhoun]

Does it jump? I feel like it's a "fat perimeter".

I kind of ran into this when I was in high school and was introduced to limits.

For me the quandary was a "stair step" shape dividing a square with length of side "s" ("stairs" connecting two opposite diagonal corners). You could increase the number of steps—they get smaller—but the total rise + run of the stairs remains the same (2s). At infinity I reasoned you had a straight, diagonal line that should have been s√2 but was also still 2s in length.

At the very least you can say that the volume enclosed approached that of a right triangle (at infinity) but the perimeter stays stubbornly the same and not that of a right triangle at all.

[arjie]

This is indeed the common way most people encounter this. The proof of the difference in the limits for the perimeter vs. the area is in the first answer to the stack overflow question in the G(^n)P: https://math.stackexchange.com/a/12907