Can a set be unbounded under discrete metric? Suppose $(X,m)$ is a metric space and $A$ is a subset of $X$. Given that $m(x,y)$ is the discrete metric, can there possibly be a set that is unbounded wrt. $m$?

If by "the discrete metric" you mean that $m(x,y)=1$, then the answer is of course not. Since the entire space is bounded by $1$.
If by "the discrete metric" you mean any metric which induces the discrete topology, then $\Bbb N=X=A$ is an example.
