Sum of r Powers is not Greater than 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:
 * $a_1^n + a_2^n + \cdots + a_r^n \le r a^n$

Proof
By definition of the $\max$ operation:


 * $\exists k \in \set {1, 2, \ldots, r}: a_k = a$

Then:
 * $\forall i \in \set {1, 2, \ldots, r}: a_i \le a_k$

Hence:

Hence the result.