Subset of Bounded Subset of Metric Space is Bounded

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Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $B$ be a bounded subset of $M$.

Let $\map \diam B$ denote the diameter of $B$.


Let $C \subseteq B$ be a subset of $B$.

Then $C$ is a bounded subset of $M$ such that:

$\map \diam C \le \map \diam B$


Proof

\(\ds \forall x, y \in B: \, \) \(\ds \map d {x, y}\) \(\le\) \(\ds \map \diam B\) Definition of Diameter of Subset of Metric Space
\(\ds \leadsto \ \ \) \(\ds \forall x, y \in C: \, \) \(\ds \map d {x, y}\) \(\le\) \(\ds \map \diam B\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds \sup \set {x, y \in C: \map d {x, y} }\) \(\le\) \(\ds \map \diam B\) Definition of Supremum
\(\ds \leadsto \ \ \) \(\ds \map \diam C\) \(\le\) \(\ds \map \diam B\) Definition of Diameter of Subset of Metric Space

$\blacksquare$


Sources