TITLE: An inequality on Hausdorff distance of union
QUESTION [1 upvotes]: Definitions: Let $X,Y$ bet sets in $\mathbb{R}^n$ then we define:
$$B(X,\epsilon):=\{r\in\mathbb{R}^n|\exists{}x\in{X},\text{ s.t. }{}d_{\mathbb{R}^n}(r,x)<\epsilon\}$$
$$d_H(X,Y):=\inf\{\epsilon>0|X\subset{}B(Y,\epsilon),Y\subset{}B(X,\epsilon)\}$$
I am trying to show that for $A,B,C,D\subset{\mathbb{R}^n}$ we have:
$$d_H(A\cup{}B,C\cup{}D)\le{}\max\{d_H(A,C),d_H(B,D)\}.$$
My attempt:
I think the following bound is clear and a good starting point:
$$d_H(A\cup{}B,C\cup{}D)\le{}\max\{\inf\{\epsilon>0|A\cup{}B\subset{}B(C\cup{}D,\epsilon)\},\inf\{\epsilon>0|C\cup{}D\subset{}B(A\cup{}B,\epsilon)\}\}.$$
I'm not sure how to proceed, a hint would be appreciated.

REPLY [1 votes]: Let $D = \max\{ d_H(A, C), d_H(B, D)\}$. For all $e >D$ we have $e > d_H(A, C)$ and $e> d_H(B, D)$. Thus by definition, 
\begin{align*}
 A \subset B (C, e)&, C \subset B(A, e), B \subset B (D, e), D \subset B(B, e).
\end{align*}
This implies $A\cup B\subset B(C\cup D, e)$ (by the first and third inclusions) and $C\cup D \subset B(A\cup B, e)$. Thus $d_H(A\cup B, C\cup D)\le e$. Since this is true for all $e >D$, the inequality is shown.
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