Let $U$ be a bounded open set in $\mathbb{R}^2$. Suppose the sets $V_m$ are an increasing sequence of closed sets such that $V_{m}\subset V_{m+1}$ and that $\cup_{n=1}^\infty V_n=U$. Then does it hold that $d_H(\partial\cup_{n=1}^k V_n,\partial U)\to0$ as $k\to \infty$, where $d_H$ is the Hausdorff distance and $\partial$ denotes the boundary of a set.
I am relatively certain I have managed to prove this in the case where the sets $V_k$ are dyadic cubes but my proof is long and cluncky. I was wondering if I could relax the conditions a bit and so obtain a shorter proof. So does the above hold in general?