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Claim: Suppose $X$ is a non-empty set and $d$ is a metric on $X$. Let $S(X)$ denote the collection of all non-empty closed bounded subsets of $X$. For each $A$ and $B$ in $S(X)$, define

$$h(A,B) = \max~ \big ( ~\sup ~\{~ dist~(b,A)~|~b \in B ~\},~ \sup ~\{~dist~(a,B)~|~a \in A~ \}~ \big )$$ Then $h$ is a metric on $S(X)$. It is called the Hausdorff metric.

Proof is as follows :

enter image description here However, I haven't been able to find an exact use for taking $S(X)$ as the set of all bounded subsets of $X$ which are also non empty and closed.

Could someone please tell me the use of taking bounded subsets here?

Thank you very much for your help in this regard.

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If you allow unbounded sets there is no reason for $h(A,B)$ to be finite. For instance, if $X=A=\mathbb{R}$ and $B=\{0\}$, then $\sup \operatorname{dist}(a,B)$ is clearly infinite.

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