Why is 0 not a limit point of this metric space in $\mathbb{R} \times \mathbb{R}$

Question

The open set $\left\lbrace \left(1/n, n \right) : n \in \mathbb{N} \right\rbrace$ under the Euclidean metric in $\mathbb {R}^2$.

Answer 1

I guess the op is talking the set $\{(1/n,n):n\in\mathbb{N}\}$. This set has no limit points in $\mathbb{R}^2$ with the Euclidean metric

Answer 2

In red the some points of the your set.