Density of Q in R .... Sohrab Theorem 2.1.38 ....

[Math Amateur]

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with an aspect of the proof of Theorem 2.1.38 ...

Theorem 2.1.38 reads as follows:https://www.physicsforums.com/attachments/7089In the above text by Sohrab, at the start of the proof, we read the following:

"We may assume that \(\displaystyle x \gt 0\). (Why) ... ... "Can someone please explain to me why we can assume that \(\displaystyle x \gt 0\)?

Peter

 

[Opalg]

Can someone please explain to me why we can assume that \(\displaystyle x \gt 0\)?Peter

For a sufficiently large integer $k$, $x+k$ will be positive. If you can find a rational number $r$ between $x+k$ and $y+k$, then $r-k$ will be a rational number between $x$ and $y$.

 

[Math Amateur]

For a sufficiently large integer $k$, $x+k$ will be positive. If you can find a rational number $r$ between $x+k$ and $y+k$, then $r-k$ will be a rational number between $x$ and $y$.

Thanks Opalg ...

Easy when you see how it works! ... :) ...

... appreciate the help ...

Peter