Intersection of a sequence of Hilbert spaces

Question

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

Let $\forall m \in \mathbb{N}: W_m = \overline{\operatorname{span}} \{v_n\}_{n \geq m}$ with the property $$ \forall m \in \mathbb{N}: v_m \notin W_{m+1} $$ Is it true that: $$ \bigcap_{m=1}^\infty W_m = \overline{span}\{u\} $$

Thanks.

Answer 1

Edited after the first answer did not work as pointed out by Matey Math:

Let $(e_n)_{n\in \mathbb N}$ be the standard basis of $\ell^2$. Then the sequence $$ v_n := e_1 + \frac{e_2 }{n} + \frac{e_{n+2}}{n^3}$$ is linearly independent. Moreover, it converges to $e_1=:u$.

However, $e_2$ is in the closure of $W_m$ for all $m$. This can be seen by looking at the sequence $n(n+1)(v_n-v_{n+1})$ which converges to $e_2$.

In conclusion, the statement does not hold.