Let $\{v_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $V$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $V_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m}$

Let $\{u_n\}_{n \in \mathbb{N}}$ be a Schauder basis of $U$ subspace of $\ell^2$ over $\mathbb{C}$ and $\forall m \in \mathbb{N}$ let $U_m = \overline{\operatorname{span}} \{u_n\}_{n \geq m}$

Under the hypothesis that $\forall m \in \mathbb{N}: V_m + U_m$ is closed, is it true that:

$$ \bigcap_{m=1}^\infty \left( V_m + U_m \right) = \{0\} $$