Let $m$ be a Lebesgue measure and $A$ a Lebesgue measurable subset of $\mathbb{R}$ with $m(A)<\infty$. Let $\varepsilon>0$. Show there exist $G$ open and $F$ closed such that $m(G-F)<\varepsilon$.
My attempt at a proof, I have only done one part so far: Since $A$ is lebesgue-measurable then $m(E)=m(E\cap A)+m(E\cap A^\complement)$ for all $E\subset\mathbb{R}$. Choose $E$ to be an open set containing $A$. Then we can write $m(E\setminus A)=m(E\cap A^\complement)=m(E)-m(E\cap A)=m(E)-m(A).$
Here I am stuck. How can I further choose $E$ so that I know the difference is going go be less than the given $\varepsilon$? If I can get this, and I set $G=E$, I have the first part done. Furthermore, is there a similar way to show the $F\subset A$ part?