Bounded Curve

Definition: Bounded Curve

A curve $\mathcal{C}$ is bounded if there exists an open ball $B$ of finite radius which completely contains it, i.e. $\mathcal{C}\subseteq B$.

INTUITION

This just means the curve is contained in a finite region of space, i.e. it does not extend infinitely in some direction. The open ball $B$ is like an $n$-dimensional sphere which completely envelopes the entire curve $\mathcal{C}$.