\begin{lemma}[Boundedness]
    Let $(X, d)$ be a metric space, and let $\varnothing \neq Y \subseteq X$. The following are equivalent:
    \begin{itemize}
        \item For every $x \in X$, there is an $R(x) > 0$ such that $Y\subseteq B(x, R)$
        \item There exists $y\in Y$ and $R$ such that $Y\subseteq B(y,R)$
        \item There is an $R> 0$ such that for any $y_1, y_2\in Y$, we have $d(y_1,y_2) < R$
    \end{itemize}
\end{lemma}