It is known that $ l^p \subseteq l^r $ if $ 1 \le p \le r < \infty $. Also, $ l_w^p $ are spaces where we consider a collection of positive weights, $ w_n > 0 $ for each $ n \in \mathbb{N} $, that given sequence $ x = (x_n) $, the weighted $p$-norm is $$ ||x||_{w,p} = \big (\sum_{n=1}^\infty w_n |x_n|^p \big )^\frac{1}{p} \text{.} $$ Prove that if $ \sum_{n=1}^\infty w_n < \infty $, then $ l^1_w \not \subseteq l^2_w $.One way to prove this is to find a subsequence $ (w_{n_k}), $ so that $ w_{n_k} \le \frac{1}{2^k} $ and consider elements which are nonzero only at the $ n_k $ coordinates.
the subsequence is probably very easy to think up, but I can just not figure it out... :(
Let $(n_k)$ be the increasing sequence of integers mentioned in the opening post and define $x_{n_k}=\left(w_{n_k}k^2\right)^{-1}$ and $x_l=0$ if $l$ is not equal to $n_k$ for some $k$. The series $\sum_{k}w_{k}\left\lvert x_{k}\right\rvert$ is convergent and $\sum_{k}w_{n_k}x_{n_k}^2$ is divergent since $w_{n_k}x_{n_k}^2\geqslant 2^k/k^4$.