Thread: elementary number theory question

If x_1, ..., x_n are n positive real numbers such that x_1*x_2* *** *x_n = 1, then prove that x_1 + x_2 + ... + x_n >= n (where >= means greater than or equal to). I'm thinking of using induction, and i know that the cases for n = 1 and n = 2 are true, but I don't know how to proceed from there. Thanks in advance for any help and hints.

Keep in mind that you are actually trying to prove the arithmetic-geometric mean inequality.

One easy way to do this is to use Lagrange multipliers, optimizing subject to the constraint .