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?

Note that
$$
\left(\frac1p\sum_{i=1}^pa_i^n\right)^{1/n}\le\left(\frac1p\sum_{i=1}^p\max_k(a_k)^n\right)^{1/n}=\max_k(a_k)
$$
and
$$
\left(\frac1p\sum_{i=1}^pa_i^n\right)^{1/n}\ge\left(\frac1p\max_k(a_k)^n\right)^{1/n}=\left(\frac1p\right)^{1/n}\max_k(a_k)
$$
Therefore,
$$
\left(\frac1p\right)^{1/n}\max_k(a_k)\le\left(\frac1p\sum_{i=1}^pa_i^n\right)^{1/n}\le\max_k(a_k)
$$
By the Squeeze Theorem, we get
$$
\lim_{n\to\infty}\left(\frac1p\sum_{i=1}^pa_i^n\right)^{1/n}=\max_k(a_k)
$$
