$ \sup \{|x_n| \mid n\in \mathbb N \}= \lim_{n\to \infty} (\lim_{p\to \infty}(\sum_{k=1}^n |x_k|^p)^{1/p}) $

I have to prove that for a bounded sequence $ {x_n}$

$$ \sup \{|x_n| \mid n\in \mathbb N \}= \lim_{n\to \infty} (\lim_{p\to \infty}(\sum_{k=1}^n |x_k|^p)^{1/p}) $$

I know that$$ \sup \{|x_n| \mid n\in \mathbb N \}= \lim_{p\to \infty}(\sum_{k=1}^\infty |x_k|^p)^{1/p}$$

if it is correct to switch the order of these limits, I can ger the desired result directly; Since:

$$ \lim_{n\to \infty} (\lim_{n\to \infty}(\sum_{k=1}^n |x_k|^p)^{1/p})=\lim_{p\to \infty} (\lim_{n\to \infty}\sum_{k=1}^n |x_k|^p)^{1/p} =\lim_{p\to \infty}(\sum_{k=1}^\infty |x_k|^p)^{1/p}=\sup \{|x_n| \mid n\in \mathbb N \}$$

But is it correct? Can we switch the order of these limits in general? or is there any condition for that?