 Solution

Because every subset of $\mathbb{N}$ is open, the Borel sigma-algebra of a discrete space isIf $f:\mathbb{N}\to \mathbb{R}$ or $f:\mathbb{N}\to [0,\infty ]$ and $B$ is a Borel subset of the codomain, then $f^{-1}(B)$ is some subset of $\mathbb{N}$, hence is Borel. Thus every such function is a Borel measurable function.