\begin{definition}[Definition:Bounded Subset of Connected Riemannian Manifold]
Let $\struct {M, g}$ be a connected Riemannian manifold.
Let $A \subseteq M$ be a subset.
Let $p,q \in M$ be points.
Let $d_g$ be the Riemannian distance.
Let $C \in \R$ be a constant.
Suppose:
:$\exists C \in \R : \forall p, q \in A : \map {d_g} {p, q} \le C$
Then $A$ is said to be '''bounded'''.
\end{definition}