Infinite decomposition of a Hilbert space

Question

Let $H$ be a Hilbert space. Suppose we have $H_{n}\oplus G_{n}=H$ for all $% n\geq 0$ so that $\left( H_{n}\right) _{n\geq 0}$ is an increasing sequence of closed subspaces and $\left( G_{n}\right) _{n\geq 0}$ is a decreasing sequence of closed subspaces. I think we don't have in general : $H=% \overline{\underset{n\geq 0}{\bigcup }H_{n}}+\underset{n\geq 0}{\bigcap }% G_{n}$. My question is do we have : $H=\overline{\underset{n\geq 0}{\bigcup }H_{n}+\underset{n\geq 0}{\bigcap }G_{n}}$ ? This statement looks simple but seems to me hard to be confirmed or even disproved with a counter-example.