Closed on Topology

Question

For $S=\{1,\frac12,\frac13,\dots,\frac1n\}$

Prove that $S$ is closed on $\mathbb R$

I thought as,

$S=\bigcup_\limits{i=1}^n \frac{1}{n}$ for any $x$ we can find $x=(a,b) \subset S$ such as $a=n+1$ and $b=1$.

So doesn't it make $S$ open?