The harmonic mean of a set of positive numbers is equal to the inverse of the average of the inverses of the numbers. The geometric mean of a set of positive numbers is equal to the nth root of the product of the numbers, where n is the size of the set. The arithmetic mean is equal to the average of the numbers.

It is known that the arithmetic mean is always greater than or equal to the geometric mean, given a set of positive numbers. But where does the harmonic mean fit in with these two other means? Is it greater, lesser, or inbetween? Prove it. Note that in this problem, a "set" allows repeated numbers.