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 exstract 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.