[5] The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
[3] The theorem admits a far-reaching generalization to a wide variety of functions (that have convexity or concavity properties of any order).
In particular, if we keep n for the RHS and choose n + 1 for the LHS we get: It is evident from this last line that a function is being sandwiched between two expressions, a common analysis technique to prove various things such as the existence of a limit, or convergence.
The remaining loose end is the question of proving that Γ(x) makes sense for all x where exists.
If, say, x > 1 then the fact that S is monotonically increasing would make S(n + 1, n) < S(n + x, n), contradicting the inequality upon which the entire proof is constructed.