Calculus – Why the Interior of Q in R is Empty

Question

I don't understand why the interior of $\mathbb{Q}$ in $\mathbb{R}$ is empty, since, for every ball with the center being a rational number, given an $\epsilon>0$, I can find an infinite sequence of rational numbers that approach this point. For example, take $\frac{1}{2}$. The sequence $\frac{1}{2}+\frac{1}{n}$ can be made as close as I want to the number $\frac{1}{2}$, therefore I can always have open balls with center $\frac{1}{2}$ such that there are rationals inside it.

Answer 1

It doesn’t matter that there are rationals inside the ball: what matters is that your open ball is not a subset of $\Bbb Q$. In order for $\Bbb Q$ to be open in $\Bbb R$, for each $q\in\Bbb Q$ there would have to be an open ball $B(q,\epsilon)$ about $q$ such that $B(q,\epsilon)\subseteq\Bbb Q$, and that is never the case: every open interval in $\Bbb R$ contains irrational numbers.