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.
(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?
(b) Now draw the graphs of
for n = 1,2,3, ... and 0 ≤ x ≤ 2π. 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 π is the same for all values of n.