Bounded set

A set that fits inside some disk of finite radius; unbounded otherwise.

A set $S$ is bounded if it fits entirely inside some disk $|z|<M$ for a large enough (but finite) $M$ -- equivalently, if the moduli $|z|$ of its points don't run off to infinity. A set that admits no such $M$ is unbounded.

Examples

The disk $|z-3|\le 5$ is bounded: taking $M=8$ , every point of the disk satisfies $|z|<M$ .
The half-plane $\operatorname{Re}(z)>0$ is unbounded -- no single disk centered at the origin contains all of it.