A closed set in a metric space is $G_\delta$
If $x\in \bigcap_n B_n$, then for each $n$, by the definition of $B_n$, there is some $x_n\in F$ with $d(x_n,x)<1/n$. Then $(x_n)$ converges to $x$, so as $F$ is closed $x\in F$.
If $x\in \bigcap_n B_n$, then for each $n$, by the definition of $B_n$, there is some $x_n\in F$ with $d(x_n,x)<1/n$. Then $(x_n)$ converges to $x$, so as $F$ is closed $x\in F$.