Exercise
Prove that every metric space $(X,d)$ admits a metric $d'$ equivalent to $d$ which makes $X$ bounded.
EDIT
Definition
Let $X$ be a non-empty set. We say the metrics $d_{1}$ and $d_{2}$ defined over $X$ are equivalent iff for every $x\in X$ and every $r > 0$:
(a) exists $s > 0$ such that $\{y\in X : d_{1}(x,y) < s\} \subset \{y\in X : d_{2}(x,y) < r\}$,
(b) exists $t > 0$ such that $\{y\in X : d_{2}(x,y) < t\} \subset \{y\in X : d_{1}(x,y) < r\}$.
My attempt
Based on the suggestion of @TheoBendit, if we take $d'(x,y) = \min\{d(x,y),1\}$, one has that for every $x\in X$ and every $r > 0$ there exists $s = \min\{r,1\}$ such that
\begin{align*}
\{y\in X : d'(x,y) < s\} \subset \{y\in X : d(x,y) < r\}.
\end{align*}
But I fail to prove the converse.
Any help is appreciated.