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!