Cauchy's Mean Theorem

Theorem
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive.

Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.

Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.

Then:
 * $A_n \ge G_n$

with equality holding :
 * $\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$

That is, all terms are equal.

Also known as
It is widely known as the Arithmetic Mean-Geometric Mean Inequality or AM-GM Inequality.

Some sources give this as Cauchy's formula.

Also see

 * Arithmetic Mean is Never Less than Harmonic Mean