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**JoachimAgrell** Let $\displaystyle (A_\lambda)_{\lambda\in\mathbb{N}}$ be a collection of open sets of real numbers. Suppose $\displaystyle F\subset\mathbb{R}$ is such that $\displaystyle \overline{F\cap A_\lambda}=F$ $\displaystyle \forall \lambda\in\mathbb{N}$.

Show that $\displaystyle \overline{F\cap\bigcap_{\lambda\in\mathbb{N}}A_\la mbda}=F$

I've managed to prove that $\displaystyle \overline{F\cap\bigcap_{\lambda\in\mathbb{N}}A_\la mbda}\subset F$.

How can i prove the other inclusion?