Cauchys_Mean_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.