The proof is that the RHS can be written as the sum of n different
sums,
(a1b1 + a2b2 + ... + an-1bn-1
+ anbn) +
(a1b2 + a2b3 + ... + an-1bn
+ anb1) +
...
(a1bn + a2b1 + ... + an-1bn-2
+ anbn-1)
The first of these n sums is at least as big as each of the others by the
rearrangement inequality, and n times the LHS, so the LHS is at least as big as
the RHS.
Related pages in this website
The AM-GM
Inequality: the Arithmetic Mean of positive numbers is always greater than
the Geometric Mean. This is proved using Jensen's Inequality.
The Cauchy-Schwarz inequality
The Triangle Inequality
Puzzles
The Quadratic Formula
