Well lets try and make neil's proof a little more rigorous. First define a function called l with its domain as the set of all real valued continuous functions on R, and its range as R. l(f) tells you the length of the graph of f between two points a and b.
We use the so called infinite norm on the domain of l i.e we define the distance between two functions f and g as:
d(f,g) = supx in R |f(x)-g(x)|
The set of all functions is of course isomorphic to a subset of the set of all curves in R2. Thus we can speak of the length of a curve using this function l(f) provided this curve is representable by a continuous function. So now I'll use the word curve with this understanding.
Now for the proof. We have above a sequence of curves, and the limit of this sequence(under the metric defined above) is the straight line between a and b(the two points involved).
The fallacy in the proof, of course, is that the proof uses the fact that:
lim(l(fn)) = l(lim(fn))
This would be true only if l was a continuous function and what this tells you is that l is not one. So mathematically speaking thats all there is here. We cant take the limit inside the function sign unless the function is continuous.
Physically speaking this is far more interesting. We would expect the length of two very similar curves to be almost equal. So this is actually an instance of a fractal. Its difficult to fit this into the usual definition of a fractal because the object is not really self-similar, but the idea is the same - If you examine an object in detail its length is not what it seems to be if you examine the same object from a distance.