Evaluating $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ where $a_1 \ge \cdots\ge a_k \ge 0$

Question

Need to find $\lim\limits_{n\to\infty}(a_1^n+\dots+a_k^n)^{1\over n}$ Where $a_1\ge\dots\ge a_k\ge 0$

I thought about Cauchy Theorem on limit $\lim\limits_{n\to\infty}\dfrac{a_1+\dots+a_n}{n}=\lim a_n$ and something like what happen if all $a_i=0$ or $a_1=\dots=a_k$, but may be something I am thinking wrong?

Maybe it is too simple but I am not getting it; please help.

Answer 1

Note that $$a_1=[a_1^n]^{1/n}\leq [a_1^n+\cdots+a_k^n]^{1/n}\leq [ka_1^n]^{1/n}=k^{1/n}a_1$$ and apply squeeze theorem.

Answer 2

Hint: The $n$-th term of our sequence is between $a_1$ and $k^{1/n}a_1$.