A simple question on Hausdorff distance

Question

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$.

Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then

$$ A_n \rightarrow A $$

in the Hausdorff distance?

Intutively this is true. I am trying to use the definition of this distance, but it's not working. Someone can give me a help to prove or disprove?

Thanks in advance

PS: for the Hausdorff distance see: http://en.wikipedia.org/wiki/Hausdorff_distance

Answer 1

This is not true. Just pick some $K$, like a closed ball, and let $A$ and $A_n$ be whatever compact subsets of $K$ you want. Then trivially $A_n\cup K\to A\cup K$.