# How would I go about proving that for any two real numbers $a,b\in\mathbb{R}$ we may find a both a rational and irrational number between them.

How would I go about proving that for any two real numbers $a,b\in\mathbb{R}$ we may find a both a rational and irrational number between them?

It can be proven by using infinite decimals but how would one prove this directly from the completeness of $\mathbb{R}$.

• You know, David, it really is polite to read the answers, upvote the helpful ones, accept the one that helped the most, and at the very least provide some commentary if you found them unhelpful. – The Count Apr 27 '17 at 19:51

You can use the Archimedean property of $\Bbb R$. Take $b \gt a$. There is some $n$ such that $\frac 2{n} \lt b-a$. There are at least two rationals with denominator $n$ between $a$ and $b$. Then the lower one plus $\frac {\sqrt 2-1}{n}$ is an irrational between $a$ and $b$
If $a$ and $b$ are both rational, then for the rational's existence, consider $\dfrac{a+b}{2}$. If either are irrational, this works as well to prove the irrational's existence. In either case, you need to be a little more clever for the other part, but as a hint: try proof by contradiction. Then what can be said about the density/completeness of the irrationals or rationals?
• We weren't given that $a,b$ are rational. – Ross Millikan Apr 26 '17 at 16:40