Notes - Metric Spaces MT23, Boundedness

What does it mean for a $Y \subseteq X$ to be bounded, where $X$ is a metric space?

$Y$ is contained in some open ball.

Prove that the following are equivalent:

$Y$ is bounded (i.e. it is contained in some open ball)
$Y$ is contained in some closed ball
The set $\{d(y _ 1, y _ 2) : y _ 1, y _ 2 \in Y\}$ is a bounded subset of $\mathbb R$.

$(1) \implies (2)$: Immediate, take the closure
$(2) \implies (3)$: We have $\exists K \in \mathbb R$, $\exists y _ 0 \in Y$ such that $\forall y \in Y$, $d(y, y _ 0) \le K$. Then $\forall y _ 1, y _ 2 \in Y$, $d(y _ 1, y _ 2) \le d(y _ 0, y _ 1) + d(y _ 0, y _ 2) \le 2K$ by triangle inequality, so the set is bounded.
$(3) \implies (2)$: We have $\exists K$ such that $\forall y _ 1, y _ 2 \in Y$, $d(y _ 1, y _ 2) \le K$. Then pick $y _ 0 \in Y$ arbitrarily. Then $\forall y \in Y$, $d(y, y _ 0) \le K$, so $Y$ is contained in $B(y _ 0, K+1)$.