Proof of A.M. ³ G.M. using integration

We like to prove :

Observe that for any x > 0 :

because, so long as the integrand’s t lies strictly increasing or decreasing between x and G , the sign G – x and of 1/t – 1/G , must be the same.

Equality sign holds iff x = G.

Now, replace x by and sum up over i = 1, 2, … , n.

\ A ³ G