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

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

Peter
 
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Peter said:
Can someone please explain to me why we can assume that $$x \gt 0$$?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 $y$.
 
Opalg said:
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
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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