Determine if the set of all eventually zero sequences is $G_\delta $ in $\ell_2$

Question

For the normed space $\left(\ell^ 2 , \|\cdot\|_2\right) $

I need to check if the following set is of type $ G_\delta $ , $F_\sigma$ :

$$E = \left\{ x \in \ell^2 : \mbox{ there exists } N \ge1 \mbox{ s.t } x_n = 0 \mbox{ for each } n \ge N \right\}.$$

I managed to prove that this set is $F_\sigma$.

I think it is not $G_\delta $ but im not sure how to prove it.

Thanks for helping

Answer 1

Completely revised and corrected HINT: Let $G=\ell^2\setminus E$. $E$ is an $F_\sigma$, so $G$ is a $G_\delta$. Thus, there are open sets $G_n$ for $n\in\Bbb N$ such that $G=\bigcap_{n\in\Bbb N}G_n$.

• Show that $G$ is dense in $\ell^2$, and conclude that each $G_n$ is dense in $\ell^2$.

Suppose that $E$ is also a $G_\delta$. Then there are open sets $U_n$ for $n\in\Bbb N$ such that $E=\bigcap_{n\in\Bbb N}U_n$.

• Show that $E$ is dense in $\ell^2$, and conclude that each $U_n$ is dense in $\ell^2$.

• Use the fact that $\{G_n:n\in\Bbb N\}\cup\{U_n:n\in\Bbb N\}$ is a countable family of dense open sets in the complete metric space $\ell^2$ to get a contradiction.