Sum of r Powers is between Power of Maximum and r times Power of Maximum

Theorem
Let $a_1, a_2, \ldots, a_r$ be non-negative real numbers.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a = \max \set {a_1, a_2, \ldots, a_r}$.

Then:
 * $a^n \le a_1^n + a_2^n + \cdots + a_r^n \le r a^n$

Proof
This proof is divided into $2$ parts: