Is it possible that the dimension of the intersection of a nested sequence of Hilbert space is 1?

Question

Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$

Let $\{h_n\}_{n \in \mathbb{N}} \in H$ be a sequence of linearly independent vectors in $H$

Let $$ V= \bigcap_{n=1}^\infty \overline{ \operatorname{ span }} \{h_m\}_{m \geq n} $$

Is it possible that $$ \dim V=1 $$

Thanks