Originally posted by Palynka
The set of points in the sequence converges to the set of points in the circle. That certainly is enough for pointwise convergence. (http://mathworld.wolfram.com/PointwiseConvergence.html)
The mistake is in assuming that all properties of the approximating sequence must all converge. This is not true (two examples posted above). Of course, because they do nt[/i] sequences for that approximation but that doesn't mean that this one doesn't converge.
It's been awhile since I've looked at sequences of functions, but I gave this some more thought whilst stuck on a plane today.
Construct a sequence of functions {f_n} as follows: we take f_1 to be the square, f_2 to be the square with "inverted" corners (yielding a cross shape), f_3 to be the cross shape with "inverted" corners, and so on*. Well, what have we got? I agree: the sequence {f_n} must converge uniformly to a circle, and not any kind of fractal-type entity. But, while I haven't checked any of your links (so I might be repeating something already mentioned), it certainly is true that not all geometrical properties of a curve are faithfully preserved in a limit process whose limit is the curve. The idea of path length is defined using a very specific kind of limit process which the limit process we're considering here (using "collapsing squares" or "corner inversions" ) does not match.
One interesting idea that occurs to me in considering how "badly" the "collapsing squares" sequence of functions misses the mark when it comes to actually "capturing" the circle is to consider this: f_1 only has 4 points in common with the circle, f_2 has 8 points, f_3 has 16 points, and in general f_n has 2^(n+1) points in common. Thus, as n gets larger the number of points the approximating path has in common with the circle is finite. Even if we take n "to infinity," given that n runs through positive integer values, we can only expect to intersect the circle at a countably infinite number of points, while the circle itself consists of
uncountably infinite points. So yes indeed, things are quite anomalous here. We cannot expect this process to faithfully hone in on the "true" circumference of the circle.
It's an interesting puzzle, but ultimately it abuses the defined means whereby arc length is computed. That is to say, the puzzle at memebase.com is essentially defining a different metric than the customary Euclidean metric we all know and love.
* To have true functions we can either deal strictly with the upper half of the picture, or else use parameterized curves f_n(t)=(x_n(t),y_n(t)).