Is the set $\{\sqrt2,2\sqrt2,3\sqrt2,…,n\sqrt2, …\}$ closed in $\mathbb{R}$

Question

Is the set $\{\sqrt2,2\sqrt2,3\sqrt2,…,n\sqrt2, …\}$ closed in $\mathbb{R}$?
My attempt:
Let $S=\{\sqrt2,2\sqrt2,3\sqrt2,…,n\sqrt2, …\}$
So $\mathbb{R}\backslash S=(-\infty,\sqrt2) \cup (
\bigcup_{k=1}^{\infty} (k\sqrt2,(k+1)\sqrt2)
)$
which is the union of a countably infinite number of open sets so $\mathbb{R}\backslash S$ is open and $S$ is closed.

Is my working flawed? Is $S$ even closed in the first place?

Answer 1

$S$ is closed and the working is not flawed. Note that any union of open sets is open, not just a countably infinite union of open sets