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**thedoctor818** So, if I have this correctly (in order to prove there exists a rational number between x & y):

Let $\displaystyle x,y \in \mathbb{R} \backepsilon x < y $ and consider $\displaystyle (x,y). $ We must find a rational number inside this interval. By the Archimedean Theorem, $\displaystyle \exists n \in \mathbb{N} \backepsilon n > \dfrac{1}{y-x}. \Rightarrow ny > nx + 1. $ We then find an $\displaystyle m \in \mathbb{Z} \backepsilon m \leq nx + 1 < m + 1. $ Now, we have that $\displaystyle m-1 \leq nx < ny. \Rightarrow x < \dfrac{m}{n} \leq x + \dfrac{1}{n} < y, $ thus proving an $\displaystyle \dfrac{m}{n} \in \mathbb{Q} \backepsilon \dfrac{m}{n} \in (x,y). $

Is that correct? If so, I am uncertain of how to prove this for $\displaystyle a \in \overline{\mathbb{Q}}. $