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.

Question

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 $\mathbb{R}$.

Answer 1

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?

Answer 2

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 $b$

Answer 3

You are trying to prove the density property.

It sometimes helps to know the name of something to look it up.

There is a proof here ( it seems fairly pointless/disingenuous just copying that proof in to this answer.)