Metric on natural numbers united with infinity

Question

can anyone give me an example for the following metric $d$?

Let $\Omega = \mathbb{N}_+ \cup \{ \infty \}$ and $d$ be a metric such that all points $n \in \mathbb{N}_+$ are isolated w.r.t. $d$ and $\lim_{n \rightarrow \infty} d(n,\infty) = 0$.

Thank you very much!

Answer 1

Hint: A rough idea, here, is to treat each $n$ like $\frac1n.$ What can we treat $\infty$ like to get the desired property?

Answer 2

All the points $n\in\mathbb{N}_+$ are isolated because there exists open sets $\mathcal{O}_{\epsilon}=\{y\in\Omega|d(x,y)<\epsilon\}$ around $x$ for some $\epsilon > 0$ such that $\mathcal{O}_{\epsilon}=\{x\}$.