Determine whether the set is closed or not Suppose $B = \{(x_n)_x \in l^2 \mid \sum_{n=1}^{\infty} |x_n| \leq 1\}$. Check if $B$ is closed in $l^2$ topology (by $l^2$ I mean sequences such that $\sum_{n=1}^{\infty} x_n^2 < \infty$).
I'm out of ideas how to approach this problem. I tried showing that any convergent sequence $B \ni y_n \rightarrow x \in l^2$ has its limit $x = (x_n)_n \in B$. The only thing I managed to prove is that $|\sum_{n=1}^{\infty} x_n| \leq 1$ but it doesn't even show that $x \in l^1$.

Suppose $s_m \in B$ and $s_m \to s$ in $l^2.$ Then $s_m(n)\to s(n)$ for each $n.$ Fatou's lemma says
$$\sum_n|s(n)| = \sum _n \liminf_{m\to \infty}\,|s_m(n)| \le \liminf_{m\to \infty}\,\sum_n |s_m(n)| \le 1 \implies s \in B.$$
