Show that $(x_n)$ is in $\ell^2$
For each $N\ge 1$, the map $$S_N:\ell^2\to\mathbb{C},\quad y\mapsto S_N(y)$$ defines a bounded linear functional on $\ell^2$, and evidently its norm $\|S_N\|=\left(\sum_{n=1}^N|x_n|^2\right)^{\frac{1}{2}}$. Since $\lim_{N\to\infty}S_N(y)$ exists for every $y\in\ell^2$, in particular $\{S_N(y)\}_{N\ge 1}$ is a bounded sequence in $\mathbb{C}$ for every $y\in\ell^2$. Then by uniform boundedness principle, $\sup_{N\ge 1}\|S_N\|<\infty$, i.e. $\|x\|<\infty$.