Andaleeb sent in this excellent solution.

The diagram shows some of the vertical lines drawn for values of x between 0 and 1 as described in the question. The lines are of height 1 unit at x = 0 and 1, of height 1/2 units at x = 1/2 , of height 1/4 units at x = k/4 and 1/8 units at x = k/8 and so on... up to
1
25

at
k
25

where k is a positive integer.
n 0 1 2 3 4 5 6 ... n n+1
Height 1 1/2 1/4 1/8
1
16


1
32


1
64

...
1
2n


1
2n+1`

Lines cut 2 1 2 4 8 16 32 ... 2n-1 2n

ruler

Thus if the height h lies in
1
2n
 > h > 1
2n+1

then the number of lines cut is given by
2 + 1 + 2 + 4 + 8 + ... + 2n-1 = 2 + 2n - 1
2 - 1
 = 2n + 1.
As n tends to infinity the height of the lines tends to 0 and the number of lines cut tends to infinity.