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## Hausdorff convergence and frontier convergence

1) Prove giving a counterexample that the Hausdorff convergence of a sequence of open sets $\Omega_n$ does not imply the Hausdorff convergence of the boundary of $\Omega_n$ to the boundary of $\Omega$.
2) Let $K_n$ be a sequence of compact sets converging in the Hausdorff distance to $K$. Prove that the set $\partial K_n$ has at least a point of accumulation and every such point $L$ satisfies $\partial K \subset L \subset K$.

Before trying the to solve this problem, the following theorems could be helpful:
a) If $K_1,K_2$ are compacts then $d^H(K_1,K_2)= \|d_{K_1}-d_{K_2}\|_{L^\infty}$, where $d_K=d(\cdot,K)$. Therefore $K_n \xrightarrow[]{H} K$ is equivalent to $d_{K_n}-d_K$ converges uniformly to $0$.

b) If $K_n$ is a sequence of compacts included in a fixed (great) compact $B$ then it exists some compact $K$ in $B$ such that there exists a sub-sequence $(K_{n_k}) \subset (K_n)$ which converges to $K$ in the Hausdorff distance.
(Hint: If you can include all compacts in 2) in a bigger compact $B$, then their boundaries will also be a sequence of compacts in $B$, and therefore their sequence will have at least one accumulation point.)