Bounded Metric Space is not necessarily Totally Bounded/Proof 1

Theorem

Let $M = \struct {A, d}$ be a bounded metric space.

Then it is not necessarily the case that $M$ is totally bounded.

Proof

Let $M = \struct {\R, d}$ be the real number line with the Euclidean metric.

Let $M' = \struct {\R, \delta}$ be the unity-bounded metric space on $M$ where $\delta$ is defined as:

$\delta = \dfrac d {1 + d}$

From Unity-Bounded Metric Space is Bounded, $M'$ is a bounded metric space.

From Unity-Bounded Metric Space on Real Number Line is not Totally Bounded, $M'$ is not a totally bounded metric space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces