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

Then:

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

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:

$\ds \forall i \in \set {1, 2, \ldots, r}: \, $

$\ds a_i$

$=$

$\ds a_k$

$\ds \leadsto \ \ $

$\ds \sum_{i \mathop = 1}^r a_i^n$

$\le$

$\ds \sum_{i \mathop = 1}^r a_k^n$

$\ds $

$=$

$\ds r a_k^n$

$\ds $

$=$

$\ds r a^n$

Definition of $a_k$

Hence the result.

$\blacksquare$