Equivalence of Definitions of Bounded Metric Space

Proof
Let $M = \struct {X, d}$ be a metric space.

Let $M' = \struct {Y, d_Y}$ be a subspace of $M$.

Definition 1 implies Definition 2
Let $M'$ be bounded according to Definition 1:


 * $\exists a \in X, K \in \R: \forall x \in Y: \map d {x, a} \le K$

Let $x, y \in Y$.

Then:

Thus:
 * $\map d {x, y} \le 2 K$

Thus, setting $r = 2 K$, $M'$ fulfils the conditions to be bounded according to Definition 2.

Definition 2 implies Definition 1
Let $M'$ be bounded according to Definition 2:


 * $\exists K \in \R: \forall x, y \in M': \map d {x, y} \le K$

Let $a = y$.

Then:
 * $\map d {x, a} \le K$

and so:
 * $\exists a \in Y, K \in \R: \forall x \in Y: \map d {x, a} \le K$

As $X \subseteq Y$ it follows by definition of subset that:
 * $\exists a \in Y \implies \exists a \in X$

and so:
 * $\exists a \in X, K \in \R: \forall x \in Y: \map d {x, a} \le K$

Thus $M'$ fulfils the conditions to be bounded according to Definition 1.

Definition 3 implies Definition 1
This follows from Boundedness of Metric Space by Open Ball.

Definition 1 implies Definition 4
This follows from Element in Bounded Metric Space has Bound.

Definition 4 implies Definition 1
This follows directly.