A finite subset $\{n_1 < n_2 < ... < n_k\} \subset N$ is called admissible if $k \leq n_1$. Let $E$ be the collection of admissible subset of $\mathbb{N}$
Define $S$ as the completion of finitely supported real sequences with respect to the norm
$$|x|_s:= \sup_{A \in E} \sum_{k \in A}|x_k|$$
where $x_k$ is the $k^{th}$ coordinate of $x$.
It's clear that $|x|_s = |x|_1$ (the $l^1$ norm) if $x$ has support in $E$. What is an example of a sequence that is in $l^2$, but not $S$?
Motivation is attempting to learn about Tsirelson's space from the book of Casazza and Shura