Every natural number n has ℵ0 larger successors. Likewise every unit fraction 1/n has ℵ0 smaller successors. Therefore we can define a function called NUF(x) describing the Number of Unit Fractions between x and 0:
NUF(x)=ℵ0 for x>0
NUF(x)=0 for x≤0
Acording to this function almost all unit fractions vanish in one point between 0 and (0,x). That means they cannot be distinguished.
But there must be a mistake, because all unit fractions are separated by finite distances, each one consisting of ℵ0 fractions and further real numbers. Therefore ℵ0⋅ℵ0 points must vanish in one point which is impossible. Can this be understood and explained?