On the interval 0 to 1 vertical lines are drawn at the points
x= k
2n

where k is an odd positive integer. The height of the lines at 0 and 1 are 1 unit, the height of the line at x=1/2 is 1/2 unit, the height of the line at x=1/4 and x=3/4 is 1/4 unit and the height of the line at
k
2n

is
1
2n

for all values of k.

Now consider the line y=h where
1
2n
 > h > 1
2n+1

. How many of the vertical lines from x=0 to x=1 does it cut?

What happens to the heights of these lines as n gets larger? What happens to the number of lines cut as n gets larger?