Consider the following sequence of sequences :
$x_{0}:=\{0,0,0,0......\}$,$x_{1}:=\{0,\frac{1}{2},0,\frac{1}{2}.......\}$ (i.e $0,\frac{1}{2}$ repeated infinitely), $x_{2}:=\{0,\frac{1}{3},\frac{2}{3},0,\frac{1}{3},\frac{2}{3}.0.......\}$ (i.e $0,\frac{1}{3},\frac{2}{3}$ repeated infinitely),........,$x_{n}:=\{0,\frac{1}{n+1},\frac{2}{n+1},\frac{3}{n+1},...\frac{n}{n+1},0,.......\}$ (i.e $0,\frac{1}{n+1},\frac{2}{n+1},\frac{3}{n+1},...\frac{n}{n+1}$ repeated infinitely)
Define $\mathcal H:=$ Set of all complex valued sequences say $\{a_{n}\}$ such that $\sum_{n=1}^{\infty}\frac{|a_{n}|^{2}}{n(n+1)} < \infty$.
Is it true $\{1,1,1,...\}$ (i.e all 1's) belongs to closed linear span of $\{x_{n}\}\subset \mathcal H$