Why this set is closed?

Question

I've read somewhere that the set $K=\{(\frac{1}{n}, n):n\in \mathbb{N}\}$ is a closed set in $\mathbb{R}^2$ and I have no idea how to prove this. Can someone help?

Answer 1

$K$ has no limit point, because distance of every distinct two members of $K$ is strictly greater than $1$. Hence the members of $K$ do not cluster anywhere.

Answer 2

Let $(x,y)\notin K$.

• If $y<1$, then the open ball around $(x,y)$ of radius $r=1-y$ is disjoint from $K$
• If $y\ge 1$ and $y\notin \Bbb N$, then the open ball of radius $r=\min\{y-\lfloor y\rfloor, \lceil y\rceil -y\}$ is disjoint form $K$
• If $y\in \Bbb N$, this means $x\ne \frac1y$ and this time we can take $r=\min\{|x-\frac1y|, 1\}$.