limits - How to prove this equation? ( find the maximum of a sequence)

I have a sequence, called $A$. It's elements are $a_1 , a_2 , \ldots , a_n$ for example: $(5, 11, 2)$

Then how to prove, that this formula results the highest value in the series?

$$\lim_{x \rightarrow \infty} \sqrt[x]{ \sum\limits_{i=1}^n a_i^x } = \max( a_1, \ldots, a_n )$$

Answer

hint: $$\text{max}(a_1,a_2,\cdots,a_n) \leq S \leq n^{\frac{1}{x}}\cdot\text{max}(a_1,a_2,\cdots,a_n)$$