calculus - Limit $lim_{n rightarrow infty} (A_1^n + ... A_k^n)^{1/n}= max{ A_1, ..., A_k}$

I have the following question. I was asked to compute the following limit: Let $A_1 ... A_k$ be positive numbers, does exist:

$$ \lim_{n \rightarrow \infty} (A_1^n + ... A_k^n)^{1/n} $$
My work:
W.L.O.G let $A_1= \max{ A_1, ..., A_k}$, so I have
$$ A_1^n \leq A_1^n + ... A_k^n \leq kA_1^n $$
so that

$$ A_1 = \lim_{n \rightarrow \infty} (A_1^n)^{1/n} \leq \lim_{n \rightarrow \infty}(A_1^n + ... A_k^n)^{1/n} \leq \lim_{n \rightarrow \infty} (kA_1^n)^{1/n} = kA_1 $$

Can I do something else to sandwich the limit?

Answer

You have made a little mistake. Correction:
$$ A_1 = \lim_{n \rightarrow \infty} (A_1^n)^{1/n} \leq \lim_{n \rightarrow \infty}(A_1^n + ... A_k^n)^{1/n} \leq \lim_{n \rightarrow \infty} (kA_1^n)^{1/n} = \lim_{n \rightarrow \infty} A_1{\color{blue}{k^{1/n}}}=A_1 $$