# Subset of Bounded Subset of Metric Space is Bounded

## 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

$\blacksquare$