Why this set is closed set?

Question

$A=\left\{n+\frac{1}{2n}|n\in\mathbb{N}\right\}$
Is this set closed?
I was reading answer form Example to show the distance between two closed sets can be 0 even if the two sets are disjoint where Nishrito mentioned above set is closed .
But On that account then $B=\left\{\frac{1}{2n}|n\in\mathbb{N}\right\}$
This set also become closed but I know which is not as $0$ is limit point of it which is not belong to that set. Any Help in this regard will be appreciated .
Thanks a lot

Answer 1

Yes, $A$ is a closed set. It follows from the fact that there is a $r>0$ such that the distance between any two distinct elements of $A$ is greater than $r$ ($\frac15$ will do, for instance). But $B$ is not closed and therefore there is no way of deducing that $B$ is closed from the fact that $A$ is closed.

Answer 2

The complement of $A$ is an infinite union of open intervals, so the complement is open which makes $A$ an open set.

Also $0$ is not a limit point of $A$, on the contrary it is an isolated point of $A.$

In fact every point of $A$ is an isolated point.