Power of Maximum is not Greater than Sum of Powers

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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:

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


Proof

By definition of the $\max$ operation:

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

Hence:

\(\ds a^n\) \(=\) \(\ds a_k^n\)
\(\ds \) \(\le\) \(\ds a_1^n + a_2^n + \cdots + a_k^n + \cdots + a_r^n\)

Hence the result.

$\blacksquare$


Sources