Limit of a sequence using Hölder's inequality Let $a_1,\ldots,a_p$ be positive real numbers. Find the limit of $$\left(\frac{a_1^n+\cdots+a_p^n}{p}\right)^{1/n}$$
My attempt:
I applied Hölder's and have obtained that this term is bounded below by $\frac{a_1+\cdots+a_p}{p}$ and since $a_1^p+\cdots+a_n^p \le (a_1+\cdots+a_p)^n$ is valid, applying log of limits technique, I get that it is bounded above by $a_1+\cdots+a_p$. However, I haven't obtained an actual limit. In fact I don't know the actual answer. I think it could be the lower bound I obtained, because that answer is validated for certain examples, like, by putting all values of $a_i$'s as a constant $k$, but I am not able to get a suitable idea to conclude that. Any help?

Let $a=\max(a_1, a_2, \dots, a_p)$. Then the limit is
$$\lim_{n\to\infty} a\left( \frac{\sum_i (a_i/a)^n}{p}\right)^{1/n}=a$$
