Suppose M is a metric space. Assume $A\subseteq M$. $A$ is bounded if $\exists R>0$ such that $\forall x,y \in A: d(x,y) \le R$.
I would like to show that the following are equivalent:
1) $A$ is bounded.
2) $A$ is contained in some closed ball.
Proof: Let $x,y\in A$. Since $\forall x,y \in A$, $d(x,y) \leq R$, it follows that if $x \in A$ then $x \in N_R(y)$. Hence A is contained in the closed neighbourhood (ball). (Note: $N_R(y)$ means closed ball (with equality))
Assume $A$ is contained in some closed ball. $A \subseteq N_r(x)$ for some $x \in M$. Hence $\forall x,y \in A: d(x,y) \le r$. Is my proof correct? What can I do to improve it?