# Bounded Metric Space is not necessarily Totally Bounded

## 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.

$\blacksquare$