Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to u$.

Is it possible extract a subsequence $\{v_{n_k}\}_{k \in \mathbb{N}}$ such that $$ \bigcap_{p=1}^\infty \overline{\operatorname{span}} \{v_{n_k}\}_{k > p} = \operatorname{span} \{u\} $$

Thanks.