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In this post on the Hausdorff distance $d_H$, the following is stated as a Theorem:

Theorem: Let $A,B,C \in H(X)$ (where $H(X)$ is the set of non-empty compact subsets of $X$). Then $d_H(A \cup B,C) = \max \{d_H(A,C),d_H(B,C)\}$

In the proof which is given as an answer to said post (last display), the inequalities

$$\tag{1}d_H(A,C),d_H(B,C) \le d_H(A \cup B,C)$$

are referred to as 'immediate'. I don't see why $(1)$ should hold -- is it even true?

If $(1)$ is not true, does the above 'theorem' even hold?

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It's not true, not $(1)$, nor the larger result.

Take, for example, $X = \Bbb{R}$, $A = [-1, 0]$, $B = [0, 1]$, and $C = [-1, 1]$. Then $A \cup B = C$, so $d_H(A \cup B, C) = 0$. However, both the theorem and $(1)$ would imply that $d_H(A, C)$ and $d_H(B, C)$ are both $0$ too, which is not the case, as $A \neq C$ and $B \neq C$ (indeed, it's easy to show that both these distances are $1$). So, the result is false in general.

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