An interval (x,y) is a set of all Real #'s, such that x < Element < y. If N is a set of intervals, such that no two of the intervals in the set intersect, then prove that N is countable.
I'm not sure how this would be countable, since the set of real numbers isn't. Anyone know how I should approach this proof? I'm guessing it should be with induction, right?
Suppose not. That is, suppose we have uncountably many disjoint intervals. By elementary properties of real numbers, each interval $(x,y)$ must contain a rational number $q$. Since the intervals are disjoint, there means there are uncountably many rational numbers, a contradiction. Therefore, there can be at most countably many such disjoint intervals.