Let $E= \{x=(\xi_{n}: \exists n_x: \xi_m = 0 \quad \forall m>n_x \}$. We define $F=\{ x =(\xi_n) \in E : \sum_{n=1}^\infty \frac{\xi_n}{n} = 0 \}$ subspace of E. Prove that this subspace is closed.
I try proving that if $x = (\xi_n)$ is the limit of a sequence $(x_n)= (\xi_1^n, \xi_2^n,...) \subset F$.
Must verified:
$\sum_{n=1}^\infty \frac{\xi_n}{n} = 0$.
$\exists n \in \mathbb{N} : \xi_n = 0 \quad \forall n>m$.
We have already proved 1). But, we don´t know how to prove 2). (We are using always $||.||_2$)
