calculus - Proving that the limit of $root n of {{a_1}^n + {a_2}^n + ... + {a_k}^n}$ is $max(a_1,...a_k)$

Prove that:

$$\lim_{n \to \infty} \root n \of {{a_1}^n + {a_2}^n + ... + {a_k}^n} = \max \left\{ {{a_1},{a_2}...{a_k}} \right\}$$

I am familiar with the theorem which says that if

$$\mathop {\lim }\limits_{n \to \infty } {{{a_n}} \over {{a_{n - 1}}}} = L$$

then,

$$\mathop {\lim }\limits_{n \to \infty } \root n \of {{a_n}} = L$$

So, I tried evaluating the expreesion:

$${{{a_1}^n + {a_2}^n + ... + {a_k}^n} \over {{a_1}^{n - 1} + {a_2}^{n - 1} + ... + {a_k}^{n - 1}}}$$

but pretty much got stuck here. Is this the right path to go?

Answer

$$\sqrt[n]{a_{\max}^n} \leq \sqrt[n]{a^n_1 + \dots + a_k^n} \leq \sqrt[n]{k \cdot a_{\max}^n}$$