> By definition, we cannot have all elements be above average: at least one has to be below.
Ludicrously pedantic nitpick: this only applies to finite sets - consider the sequence 1,1/2,1/3,1/4,... (the harmonic series). The average (mean, median, mode[0]) is (depending on how pedantic you want to be) either 0 or 0+ε[1], but in any case strictly less than any positive real number, while every element of the sequence is a positive real number.
So it's not true by definition; it's a consequence of the basic sanity constraints that you're working with.
0: Strictly speaking mode only applies to continuous ditributions (ie, with a continuous probability density function) or fully discrete distibutions (eg heads vs tails), but 0 is the only (real number) x such that for any sufficiently small positive distance δ, the number of elements in x±δ is strictly greater than the number in x±2δ but not in x±δ (namely, all but a finite number of the inifitely many elements in x±2δ are also in x±δ).
1: Where ε is some surreal number[2] strictly less than any positive real number, but not necessarily 1/ω specifically.
2: https://en.wikipedia.org/wiki/Surreal_number