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In Rudin's Principles of Mathematical Analysis there's the following proof that Q is dense in R.

Theorem: If [itex] x,y\in \mathbb{R}[/itex] and [itex]x < y [/itex] there exists a [itex]p \in \mathbb{Q} [/itex] such that [itex] x<p<y.[/itex]

Proof: Since [itex]x<y[/itex], we have [itex]y-x>0[/itex]. It follow from the Archimedian property that there is a positive integer [itex]n[/itex] such that

[tex]n(y-x)>1.[/tex]

We again apply the Archimedian property to find positive integers [itex]m_1[/itex] a,d [itex]m_2[/itex] such that [itex]m_1>nx[/itex] and [itex]m_2>-nx[/itex]. Then

[tex]-m_2<nx<m_1.[/tex]

Hence there is an integer [itex]m[/itex] (with [itex]-m_2\leq m\leq m_1[/itex]) such that

[tex]m-1\leq nx < m.[/tex]

We combine the inequalities to get

[tex]nx < m \leq 1+nx < ny.[/tex]

n is positive so

[tex]x < \frac{m}{n} < y.[/tex]

Which proves that [itex]\mathbb{Q}[/itex] is dense in [itex]\mathbb{R}[/itex].

How one concludes that the m in the red bit exists is what's troubling me.

Thanks

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# Q is dense in R question about proof

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