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 $v_n \to v_0$.

I would like to know if exist a subsequence $\{v_{n_k}\}_{k \in \mathbb{N}}$ such that for each fixed $p \in \mathbb{N}$: $$ v_{n_p} \notin \overline{span} \{v_{n_k}\}_{k > p} $$

Thanks.