Is the intersection of the following closed and open set closed? Generally? Ok, I have been informed that the below lemma is incorrect. I needed it to prove the following statement. Could someone else provide a proof?
Statement: If m(E) is finite, there exists a compact set K with $K\subset E$ and $m(E-K)\le \epsilon$. 
m(E) is the Lebesgue measure of set E.
Thanks!
Lemma: Let F be a closed subset of $\mathbb{R^d}$. Let $B_n(0)$ be a ball of radius n centered at the origin. Let $K_n=F\cap B_n$. Show $K_n$ is compact.

Take $K_n:=\overline{F\cap B_n}$, then $$\lim_{n\to\infty}\lambda(K_n)=\lambda(F),$$
where $\lambda$ denotes the Lebesgue measure. It should be easy to conclude from here.
