# a measurable set intersect a compact set

A set $E\subset \mathbb{R}$ is measurable if given $\epsilon >0$ there is an open set $G$ and a closed set $F$ such that $F\subseteq E \subseteq G$ and $m(G-F)<\epsilon$, where $m$ is the outer measure. I'm trying to prove that if $E$ is a measurable set then $E\cap K$ is measurable for all compact $K$ (in $\mathbb{R}$). This is what I have so far:

Let $E$ be a measurable set and $K$ be a compact set. Since $K$ is compact it must be closed and bounded and all closed sets are measurable. Now given $\epsilon > 0$, there are open sets $G_K, G_E$ and closed sets $F_K, F_E$ such that $F_K\subseteq K \subseteq G_K$, $F_E\subseteq E \subseteq G_E$, $m(G_K-F_K)<\epsilon$ and $m(G_E-F_E)<\epsilon$. We then have $(F_E\cap F_K)\subseteq (E\cap K)\subseteq (G_E\cap G_K)$, where $(F_E\cap F_K)$ is closed and $(G_E\cap G_K)$ is open. Now I'm having trouble showing that $m((G_E\cap G_K)-(F_E\cap F_K))< \epsilon$. Any suggestions? Is there a way to write $(G_E\cap G_K)-(F_E\cap F_K)$ as the union of (nice) disjoint sets? I realize that I should probably use $\frac{\epsilon}{2}$ up above so everything works correctly.

But all compact sets are measurable, so $E\cap K$ is the intersection of two measurable sets which we should know is measurable?