Proving $\Bbb R^n$ can be covered with a countable number of special sets.
Solution 1:
Your attempt is too vague. Let us show that the countably many sets $S_\epsilon(q)$ with $q \in \mathbb Q^n$ cover $\mathbb R^n$.
Since $S$ is open and $0 \in S$, there exists $r > 0$ such that $B_r(0) \subset S$.
Consider $x \in \mathbb R^n$. Since $\mathbb Q^n$ is dense in $\mathbb R^n$, we find $q \in \mathbb Q^n$ such that $\lVert x - q \rVert < \epsilon r$. Then $y = \frac{1}{\epsilon}(x - q)$ has the property $\lVert y \rVert < r$ and thus $y \in S$. This shows that $$x = q + \epsilon y \in S_\epsilon(q) .$$