# Using a specific Lemma, how to prove that between any two distinct rational numbers there exists an irrational number? Lemma: If m/n and r/s are rational, with r/s ≠ 0, then m/n + r/s xx sqrt 2 is irrational.

Using a specific Lemma, how to prove that between any two distinct rational numbers there exists an irrational number?
Lemma: If $m/n$ and $r/s$ are rational, with $r/s\ne 0$, then $r/s\ne 0$ is irrational.
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martinmommy26nv8
The lemma simply says that if $p$ and $q$ are rational, and $q\ne 0$, then $p+q\sqrt{2}$ is irrational; it doesn’t matter which rational numbers $p$ and $q$ are (as long as $q\ne 0$). Now take $p=\frac{m}{n}$ and $q=\frac{r}{s}-\frac{m}{n}$; these are rational, and by hypothesis $\frac{m}{n}<\frac{r}{s}$, so $q\ne 0$, so the lemma applies to let us conclude that $p+q\sqrt{2}=\frac{m}{n}+\sqrt{2}\left(\frac{r}{s}-\frac{m}{n}\right)$ is irrational.