Does one of these open sets contain this open set? Let $V$ be a bounded open set  and $(F_{n})$ a sequence of increasing closed sets (so $F_{n}\subset F_{n+1}$ for all $n$). Suppose we have for all $n$, 
$$V\subset\bigcup_{n\geq1} \overset{\circ}F_{n}$$
and 
$$\partial V\subset \bigcup_{k\geq1}\bigcap_{n\geq k}\partial F_{n}.$$
Here, all sets are in  $\mathbb{R}^{n}$ with $n\geq1$, the circle on the top of a set means its interior, and $\partial$ indicates the boundary. Can we conclude that there exits $n_{0}$ such that $F_{n_{0}}$ contains $V$?

P.S. The poster forgot to add the condition "V is bounded" at first.
I don't think so and the counter example is simple.
Assume $V=\mathbb{R}$ and $F_n=(-\infty,n]$
$$\mathbb{R}=\bigcup_{n\geq1}(-\infty,n)$$
and
$$\partial V=\emptyset$$
And I don't think so even though $V$ is bounded.
Assume $V=(0,1)$ and $F_n=\{0\}\cup[\frac{1}{n+2},1-\frac{1}{n+2}]\cup\{1\}$. 
$$\overset{\circ}F_n=(\frac{1}{n+2},1-\frac{1}{n+2})\quad \Rightarrow\quad V\subset \bigcup_{n\geq1}\overset{\circ}F_n $$
$$\partial V=\{0,1\}\quad \&\quad \partial F_n=\{0,1,\frac{1}{n+2},1-\frac{1}{n+2}\}$$
$$\Rightarrow  \partial V\subset\bigcup_{k\geq1}\bigcap_{n\geq k}\partial F_n $$
