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

In summary, in Chapter 2 of "Basic Real Analysis" by Houshang H. Sohrab, we encounter Theorem 2.1.38 which states that for any two real numbers $x$ and $y$, there exists a rational number between them. In the proof of this theorem, it is stated that we can assume $x>0$. This is because for a sufficiently large integer $k$, $x+k$ will be positive, and if we can find a rational number $r$ between $x+k$ and $y+k$, then $r-k$ will be a rational number between $x$ and $y$. This assumption simplifies the proof and makes it easier to understand.
  • #1
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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 \(\displaystyle x \gt 0\). (Why) ... ... "Can someone please explain to me why we can assume that \(\displaystyle x \gt 0\)?

Peter
 
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  • #2
Peter said:
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 $y$.
 
  • #3
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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FAQ: Density of Q in R .... Sohrab Theorem 2.1.38 ....

1. What is the meaning of "density" in the context of Sohrab Theorem 2.1.38?

In this context, "density" refers to the measure of how closely packed the elements of set Q are within set R. It is a way to quantify the distribution of elements in a set.

2. How is density of Q in R calculated?

The density of Q in R is calculated by dividing the number of elements in Q by the total number of elements in R. This will give a decimal value between 0 and 1, where a value of 1 indicates that all elements in R are also in Q.

3. What is the significance of Sohrab Theorem 2.1.38?

Sohrab Theorem 2.1.38 is significant because it provides a way to mathematically determine the density of a set within another set. This can be useful in various fields such as statistics, physics, and chemistry where understanding the distribution of elements is important.

4. Is there a relationship between density and volume?

Yes, there is a relationship between density and volume. Density is defined as mass per unit volume. This means that as the volume of a set increases, the density will decrease if the mass or number of elements stays constant. Similarly, if the volume decreases, the density will increase.

5. Can density be negative?

No, density cannot be negative. It is a ratio of two positive values and will always result in a positive value between 0 and 1. A density of 0 indicates that there are no elements of Q in R, and a density of 1 indicates that all elements in R are also in Q.

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