Let $X$ be a locally compact Hausdorff space and let $\mu$ be a finite Borel measure on $(X,\mathcal{B}_X)$. Suppose that $\mu$ is inner regular on open sets and that for all $A \in \mathcal{B}_X$ and $\varepsilon > 0$, there exists compact $K \subset A$ such that \begin{align*} B \in \mathcal{B}_X \text{ with } K \subset &B \subset A \implies |\mu(A)-\mu(B)|< \varepsilon \end{align*} Prove that $\mu$ is outer regular on all Borel sets.
I'm really stuck here. Is this a matter of using inner regularity on the open sets and taking complements? Or do I just hack through as if I didn't know that $\mu$ was inner regular on open sets?
Any help would be great! :)