>>12996399the sequence of curves themselves, the sequence of their perimeters, and the sequence of the areas they enclose, are three different objects. a priori there's no reason to assume that their limits are related in any way. it would be like saying "1/n > 0 therefore lim 1/n > 0" which is obviously false.
there DOES exist a theorem in math which says something like:
>if a sequence of curves c_n converge to some limiting curve c, then the areas enclosed by c_n converge to the area enclosed by c.the jagged curves in pic related do in fact converge to a circle, and we can rightfully deduce that the areas enclosed by the jagged curves converge to pi. on the other hand, there DOES NOT exist a theorem which would say
>if a sequence of curves c_n converge to a curve c, then the sequence of perimeters of c_n converge to the perimeter of cit's a non-sequitur. this picture, together with
>>12995177, is actually a counter example. the curves converge to a circle, but the sequence of perimeters is the constant sequence (4,4,4,..) which doesn't converge to 2pi. in order for this principle to hold, further conditions must be imposed on c_n, namely we must assume that also the tangent vectors of c_n converge to tangent vectors of c. this is exactly the condition which prohibits the "jaggedness".
