The problem with

Sand Jack's `

proof' is that it improperly uses

mathematical induction. We have

- The length of figure 0 (Sand Jack's first figure) is 2.
- If the length of figure
*k* is 2, then the length of figure *k*+1 is 2.

From this, mathematical induction allows us to establish that the length of figure

*n* is 2, for any natural number

*n*. However,

induction does not allow us to make such a claim for the limit. Sand Jack's proof is

parallel to the following, also incorrect, one:

Consider the sequence 1, 1/2, . . ., 1/*n*, . . . . Clearly, each element of this sequence is greater than zero. Hence the limit of this sequence (that is, zero) is greater than zero. Therefore, zero is greater than itself.

I have heard Sand Jack's proof attacked with ``the limit of the sequence of figures is not a straight line''. However, the limit *is* a straight line, for a reasonable definition of `limit': for each figure *n*, let f_*n* be the natural parametrisation of that figure. Then the sequence { f_*n* } converges (uniformly) to the natural parametrisation of the diagonal line.

To reiterate: after any finite number of steps, the

length is indeed 2, but the figure is not a straight line. In the limit, the figure does become a

straight line (not a

fractal as

Dhericean supposes), but the length is not 2.

Dhericean and

evan927 state the first part of this, but the

second is just as important.