Challenge Level

On the interval 0 to 1 vertical lines are drawn at the points
$x={k\over 2^n}$ 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
$x={1\over 2}$ is ${1\over 2}$ unit, the height of the line at
$x={1\over 4}$ and $x={3\over 4}$ is ${1\over 4}$ unit and the
height of the line at ${k\over 2^n}$ is ${1\over 2^n}$ for all
values of $k$.

Now consider the line $y=h$ where ${1\over 2^n} > h > {1\over 2^{n+1}}$ . 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?

Now consider the line $y=h$ where ${1\over 2^n} > h > {1\over 2^{n+1}}$ . 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?