Limit in a sequence of Hilbert spaces

Question

Let $H$ be an complex infinite dimensional Hilbert space

Let $\{H_n\}_{n \in \Bbb N}$ be a sequence of subspaces of $H$ such that $\cap_{n=1}^\infty H_n = H_0$, where $H_0$ is a one dimensional subspace of $H$ and such that $H_{n+1}\subsetneq H_n$

Let, for each $n \in \Bbb N$, be $v_n \in H_n$.

My questions are: is $\{v_n\}$ convergent ? if it is convergent, is $v_n \to v_0 \in H_0$ ?

Thanks for any suggestion