Awwh, come on, folks. There are 12 (yes, 12) proofs of the GM-AM Inequality in the classic little book on inequalities by Beckenbach and Bellman.
Here is an excerpt from a reader’s review of the book at Amazon:
An Introduction to Inequalities is an unexpectedly delightful book.
Relatively brief, only 129 pages, this publication of The Mathematical
Association of America, requires no more than basic high school
mathematics. Nonetheless, I am convinced that Edwin Beckenbach's and
Richard Bellman's systematic study of inequalities would interest most
students in an early calculus course.
Some classical inequalities were familiar, like the arithmetic mean -
geometric mean inequality and the Cauchy inequality (two-dimensional
version). But others like the n-dimensional version of the Cauchy
inequality (along with the Cauchy-Lagrange identity), the Hölder
inequality, and the Minkowski inequality were new to me. What I
found most surprising was how these classical inequalities were so
interrelated, and how some can be considered generalizations of others.
Beckenbach and Bellman introduce clever substitutions to transform one
inequality expression into another.
(credit: The reviewer quoted is Michael Wischmeyer, of Houston, Texas.)
Here is the link.
There is an old joke that says that sometimes a couple of months spent in the laboratory can save a couple of hours spent in the library.
In other words, let’s refrain from re-inventing the wheel, except purely as an exercise.