In both parts of this question we consider the limiting case of a
  process which is repeated infinitely often and things are not
  what they might seem to be." <br />
  <img alt="steps" src="/content/03/09/15plus2/steps.gif" /> <br />
  (a) In a square $ABCD$ with sides of length 1 unit a path is
  drawn from $A$ to the opposite corner $C$ so that the steps in
  the path are all parallel to $AB$ or parallel to $BC$ and not
  necessarily equal steps. Draw paths of this sort with 5 steps, 10
  steps, 20 steps ... Find the total length of the path in each
  case. What would the length of such a path be if it had 1000
  steps? What about the length of the path with 1 million steps? Is
  there anything surprising about this result? Suppose you keep
  increasing the number of steps in paths from $A$ to $C$ of this
  sort, putting in more and more and more steps. What can you say
  about the total length of the path?<br />
  <br />
  (b) Now draw the graphs of $y={1\over 2^n}\sin 2^nx$ for $n=
  1,2,3, ...$ and $0\leq x \leq 2\pi$. As $n$ tends to infinity the
  graphs oscillate more and more and get closer and closer to the
  $x$ axis. Prove that the length of the curve from $x=0$ to $x=2
  \pi$ is the same for all values of $n$. <br />