**Lemma** If $r$ is a real number, then for any $\epsilon>0$, there exists a rational number $x$ such that $r-\epsilon < x < r+ \epsilon$ **Proof** Take $r \in \mathbb{R}$. There is only something to prove if $r$ is irrational. Take any $\epsilon>0$ and consider $a = r - \epsilon$, $b=r+\epsilon$. By [[Existence of a rational in any closed real interval]], there is a rational number $x$ such that $r-\epsilon = a < x < b = r+\epsilon$