Two real discrete sets with non discrete union

Question

I need to find an example of two discrete sets $X,Y\cap \mathbb{R}$ such that $X\cup Y$ is not discrete. I would like to share my attempts, but I'm not seeing any way to think.

I need some tips, I don't want the example explicit

Answer 1

Hint: Prove that the set $X:=\{\frac1n: n=1,2,\ldots\}$ is discrete. Now choose set $Y$ somehow...

Answer 2

Hint: If $X\cup Y$ is not discete, then some point $z\in X\cup Y$ is a limit point of $X\cup Y$. Let's say $z\in X$. Every neighborhood of $z$ contains a point of $X\cup Y$. However, since $X$ is discrete, there must exist an $r>0$ for which the ball of radius $r$ centered at $z$ contains no other points of $X$. This means that for all $n$ such that $1/n<r$, there is a point $(y_n)\in Y$ whose distance to $z$ is less than $1/n$, so $y_n$ converges to $z\in X$. This means $z\not \in Y$, because $Y$ is discrete.

Try to concoct two sets which fit the above description.