Let 'A' be a set of open lines on the real axis, so that every two lines in this group are disjoint. Prove that |A| is lower or equal to 'Alef 0' (the cardinal number of the Natural numbers - |N|='Alef 0' )

Clue: Think of a finite line - how many lines that belong to A can be have the length equal or bigger than (1/n) in this line? (the length of the line (a,b) is* )a-b

*I believe this is a mistake, and they should have written b-a (considering that b>a)

*In case my way of explaining it in English was not quite clear - an open line can be, for instance (0,3) meaning that 0<x<3 for every x in the line. Every two lines in this group are disjoint, means that they have no common elements.

Thank you very much!