Limit of norm 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 Hilbert subspaces of $H$ such that $H_{n+1}\subsetneq H_n$ and $\cap_{n=1}^\infty H_n=H_0$ where $H_0$ is one dimensional subspace of $H$

Let, for each $n \in \Bbb N$, be $v_n \in H_n$ such that $v_n \notin H_0$ and $\lVert v_{n+1} \rVert \le \lVert v_n \rVert$

Is it true that $\lim_{n \to \infty} \lVert v_n \rVert = 0$

Thanks for any suggestion

Answer 1

This is not true. For any such construction we can always find $v_n\in H_n\setminus H_{n+1}$ which then is also clearly not in $H_0$. Afterwards we can rescale it so that it has norm $1+\frac 1n$.