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