In Mathematics Magazine 28(1954/55), 21-46, Richard Bellman presents a proof for the theorem which says that the geometric mean of $n$ numbers is always not greater than the arithmetic mean: the proof runs by showing that this is the case for $n=2^k$ and then by going backwards to "fill the blanks".
I am sure this is not the first case in which induction is used backwards, from a larger to a smaller number, but I cannot find an earlier example, or at least an early statement of that proof. Could someone help me?