How to prove that $\bar{\mathbb{Q}}=\mathbb{R}$?

Question

How to proof that $\bar{\mathbb{Q}}=\mathbb{R}$, where $\bar{\mathbb{Q}}=\mathbb{Q}\cup\mathbb{Q}^{\prime}$ and $\mathbb{Q}^{\prime}$ are the limit points of $\mathbb{Q}$?

Answer 1

Hint: Show that $\Bbb Q$ is dense in $\Bbb R$ by showing that any basic open set contains a rational point. The basic open sets are sets of the form: $\{x-\epsilon<y<x+\epsilon\}$ as $\epsilon$ and $x$ range over all real numbers. Note that this is the same as saying that a basic open set is an interval of the form $(a,b)$ for $a<b \in \Bbb R$.