# Is the following result on the Hausdorff distance even true?

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?

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.