Proof completenes of $ \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \}$

Question

Let $(s_n)_{n\in\mathbb{N}} \subseteq \mathbb{R}$ such that for all $n$: $0 < s_n \leq \frac{1}{n} $. Let $p>1$.

How to show that the space of sequences

$ l^p_s := \{x \in \mathbb{C}^\mathbb{N}\ |\ \sum_{n=1}^\infty s_n |x_n|^p < \infty \} $ is complete in respect to the norm $ ||x||:= (\sum_{n=1}^\infty s_n |x_n|^p)^{\frac{1}{p} }$.

I tried to fix a $k\in\mathbb{N}$ and show that $(x_n(k))_{n\in\mathbb{N}}$ is a Cauchy Sequence, but it doesn't work.

Answer 1

Use the fact that $\ell^{p}(\mathbb C ^{\mathbb N})$ is complete and the map $(x_n) \to (s_n^{1/p}x_n)$ is an isometric isomorphism. The condition $s_n \leq \frac 1 n$ plays no role in this. Alternatively, you can imitate the proof of completeness of $\ell^{p}(\mathbb C ^{\mathbb N})$.