Limit of Hölder Mean as Exponent tends to Infinity

Theorem
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.

For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.

Then:
 * $\ds \lim_{p \mathop \to +\infty} \map {M_p} {x_1, x_2, \ldots, x_n} = \max \set {x_1, x_2, \ldots, x_n}$

Proof
Let $p \in \R$ such that $p \ne 0$.

Let it be assumed (or arranged) that:
 * $x_1 \ge x_2 \ge \cdots \ge x_n$

Then: