Problem 1

Let \(\{x_n\}\) be a sequence of positive real number that converges to 0. For all \(n\in \mathbb{N}\), define \[a_n=\left(\sum\limits_{k=1}^{n}x_k^n\right)^{\frac{1}{n}}.\] Prove that the sequence \(\{a_n\}\) converges and find its limit!