Yeah, Cauchy's proof of the means inequality uses powers of 2 and stuff, but it's rather involved and long. I propose the following much shorter and, imo, simpler and more elegant:
Next, prove the lemma by induction: for n = 1 it's clear, so we can assume it's true for n-1 and show for n. Now, if all the numbers are equal then the inequality is trivial, so
we may assume the numbers are in increasing order, and thus (attention: if not all the numbers are equal then there must be numbers above and below 1).
The inductive hypothesis tells us that (why?) , so: