The point is this. If you give me any partition of into subintervals, let's just call them , we find that . Why? Well, let's look at something. These intervals are presumably non-degenerate (not one point) and so in any of these intervals there are both irrational and rational numbers. So, for any subinterval since it must contain an irrational. So, . But, why does this last sum equal one? The way I wrote it should give you a clue. Start with that .