Bounded Metric Space is not necessarily Totally Bounded

Theorem
Let $M = \left({A, d}\right)$ be a bounded metric space.

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

Proof
Let $M = \left({\R, d}\right)$ be the real number line with the Euclidean metric.

Let $M' = \left({\R, \delta}\right)$ 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.