If you slice anything into N pieces of all different sizes what is the average size of the pieces?
avg = (sum of piece sizes when expressed as a fraction of the total equals 1) / N
In other words:
avg = (x/N + y/N + z/N + ... ) / (x+y+z+...)
avg = (x+y+z+...) / N / (x+y+z+...) = 1/N
An infinity of pieces not included, but as Feynman pointed out, real objects can't be sliced up into an infinite number of pieces.