Let $ a_1, \ldots, a_p $ be fixed positive real numbers. Let us consider the sequences, $$s_n=\dfrac{a_1^n+a_2^n+\cdots +a_p^n}{p}\ \text{and}\ x_n=\sqrt[n]{s_n},\quad n\in\mathbb N $$ Show that the sequence $ \{x_n \} $ is monotonic increasing.
I have tried to demo using the following but have come to nothing, $$\left (\sum_{k=1}^n a_k^p \right )^2\leq \sum_{k=1}^n a_k^{p+q}\sum_{k=1}^n a_k^{p-q},\ \forall p,q\in\mathbb R, a_1,\ldots,a_n\in\mathbb R^+ $$
Let $n <m$. Then $ \sum\limits_{i=1}^{p} |a_i|^{n} \leq (\sum\limits_{i=1}^{p} |a_i|^{m})^{n/m} (\sum\limits_{i=1}^{p} 1)^{1-\frac n m}$ by holder's inequality with exponents $p=\frac m n$ and $q=\frac m {m-n}$. Just rise both sides to power $1/n$ to finish.