That's irrelevant. + The limit has all the points on the line. + Uncountable sets can be of length zero + There are families of curves where at no stage do a

That's irrelevant. + The limit has all the points on the line. + Uncountable sets can be of length zero + There are families of curves where at no stage do any of the points lie on the circle, and yet the limit is the circle, and the limit of the length is the length of the circle. In short, your comment isn't really explaining anything. ADDED IN EDIT: Whoa - cool - a downvote! Please, let me know where I'm wrong. I thought I'd explained clearly why the comment was wrong, and I have explained elsewhere more about what's really going on. No doubt you, the downvoter, have moved on now, but whoever you are, I'd love more information as to what you think I've got wrong. Thanks.