Does every non-compact bounded metric space support an equivalent metric in which it is unbounded?

Question

Consider $X$ be an infinite set. Let $d$ be a non compact bounded metric on $X$. Can we define an unbounded metric $d'$ on $X$ such that both the metric spaces $(X,d)$ and $(X,d')$ give the same topology?

^{(Since compact metric spaces are bounded, the assumption that $X$ is not compact is necessary)}

Answer 1

This follows from this answer: a metrisable space is compact iff every compatible metric is bounded.

The proof in particular shows that if $(X,d)$ is non-compact, there is an equivalent metric $d'$ such that $(X,d')$ is unbounded and the construction is pretty explicit.