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I find this explanation intuitive and visual but still got confused in rigorous proof. thanks for helping fill in the blanks. Can someone help me to fill in the blanks for the last proof? (Q is dense in R)

http://web.math.ucsb.edu/~helena/teaching/math117/density.pdf

The first blank I filled with: $\forall y\in\mathbb{R}$, the second blank by $n>\frac{1}{y-x}$, third by $1+nx<ny$

Chinnapparaj R
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Myshkin
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1 Answers1

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Since we are in the case $x>0$, $\underline{nx>0}$ and there exists $m\in \Bbb{N}$ such that $$m−1 \leq \underline{nx} <m$$(The proof that such an $m$ exists uses the well-ordering property of $\Bbb{N}$)

Then,$$ny> \underline{1+nx \geq m}$$Thus $$nx<m<ny$$ It then follows that the rational number $r=\frac{m}{n}$ satisfies $x<r<y$ .

Chinnapparaj R
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