Boundedness in Metric Spaces

Definition: Bounded Subset

Let $(M,d)$ be a metric space with the metric topology induced on it by $d$.

A subset $S\subseteq M$ is bounded if and only if there exists some open ball $B$ such that $S\subseteq B$.

Theorem: Equivalent Definition

Let $(M,d)$ be a metric space.

A subset $S\subseteq M$ is bounded if and only if there exists some $r\in {\mathbb{R}}_{\ge 0}$ such that $d(x,y)<r$ for all $x,y\in M$.

PROOF

TODO