If $m^{*}(E)<\infty$, then $E$ is measurable iff $m^{*}((a,b)\cap E)+m^{*}((a,b)\sim E)=b-a$ for all $a<b$.

Question

Question

Let E have finite outer measure. Show that E is measurable if and only if for each open, bounded interval (a,b), b - a =$ m^*((a,b) \cap E) + m^*((a,b) $~ E)

I would like to prove "if" part. My proof is as follows

Since E have finite outer measure, there exists an open set (a,b) containing E.
Let $k \in \mathbb{N}$. Then E $\subset$ $(a-\frac{1}{2k}, b+ \frac{1}{2k}) = I_k$.
By the assumption, $\frac{1}{k}$ = $m^*$($I_k \cap E) $ + $m^*$($I_k$ ~ E) = m*(E) + m*($I_k$ ~ E).
Thus m*($I_k$ ~ E) = $\frac{1}{k}$ - $m^*(E)$ < $\frac{1}{k}$ < $\epsilon$ by Archemedian Property.

Is my proof correct?