-3
$\begingroup$

i need somebody review my proof of this exercise and correct me, pls.

Prove exist $x\in\mathbb{Q}, r_{1},r_{2}\in\mathbb{I}$ such that $r_{1}

We know $-\sqrt{2}<1<\sqrt{2}$, Let $p,q\in\mathbb{Z}$ then:

Case 1: $\text{p,q>0}$

-$\sqrt{2}p

Case 2: $p>0$,$q<0$

$-\sqrt{2}p\frac{p}{q}>\sqrt{2}\frac{p}{q}$

Then exist $x=\frac{p}{q}$ such that $r_{1}=-\sqrt{2}\frac{p}{q}>x=\frac{p}{q}>\sqrt{2}\frac{p}{q}=r_{2}$

  • 0
    irrational number @OpenBall – 2017-01-21
  • 0
    What are you trying to prove? $\forall r_1$r_1,r_2 \in \mathbb{R} \setminus \mathbb{Q}, \exists x \in \mathbb{Q}$ such that $r_1< x$r \in \mathbb{R} \setminus \mathbb{Q}$ as $r=\sqrt{2} \frac{p}{q}$. – 2017-01-21
  • 1
    Your question is worded very poorly and it's hard to tell what it means. Is the problem "for all $x\in\Bbb Q$, there exists $r_1,r_2$..."? Or is it "for all $r_1,r_2\in\Bbb I$, there exists $x\in\Bbb Q$ such that ..."? – 2017-01-21

1 Answers 1

2

Your question is badly phrased. I think you wanted to say, for all $r_1,r_2\in\mathbb{R}\setminus\mathbb{Q}$ with $r_1

Pick $q\in\mathbb{Z}_{>0}$ such that $q\left(r_2-r_1\right)\geq 1$ (which exists by the Archimedean Property). By the Well Ordering Principle, let $p$ be the smallest integer with $p\geq qr_1$. Then, show that $qr_1