This morning I read the sentence "We proceed by induction" and thought, "well I guess I'm in for some insight-free symbol twiddling"
That's probably unfair. For me it's hard enough to write any proof, of any proposition, much less one that makes you feel you "understand" "why" it's true.
Yes -- proof was not super helpful to me.
The theorem was: The arithmetic mean of a collection of positive real numbers is greater than the geometric mean.
I think a good proof ought to start with how the two means are related via the exp function.
exp(arithmean(a))
= e^((a1 + a2 + ... + an)/n)
= (e^a1 * e^a2 * ... * e^an)^(1/n)
= geomean(exp(a))
Jason Orendorff