Math Amateur
Gold Member
MHB
- 3,920
- 48
I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...
I am currently focused on Section 1.5: Constructing the Rational Numbers ...
I need help with Exercise 1.5.9 (3) ...Exercise 1.5.9 reads as follows:
View attachment 7023We are at the point in Bloch's book where he has just defined/constructed the rational numbers, having previously defined/constructed the natural numbers and the integers ... so (I imagine) at this point we cannot assume the existence of the real numbers.
Basically Bloch has defined/constructed the rational numbers as a set of equivalence classes on $$\mathbb{Z} \times \mathbb{Z}^*$$ and then has proved the usual fundamental algebraic properties of the rationals ...Now ... we wish to prove that for $$r, s \in \mathbb{Q}$$ where $$r \gt 0$$ and $$s \gt 0$$ that:
If $$r^2 \lt s$$ then there is some $$k \in \mathbb{N}$$ such that $$( r + \frac{1}{k} )^2 \lt s$$ ... ...
Solution Strategy
Prove that there exists a $$k \in \mathbb{N}$$ such that $$( r + \frac{1}{k} )^2 \lt s$$ ... BUT ... without in the proof involving real numbers like $$\sqrt{2}$$ because we have only defined/constructed $$\mathbb{N}, \mathbb{Z}$$, and $$\mathbb{Q}$$ ... so I am assuming that we cannot take the square root of the relation $$( r + \frac{1}{k} )^2 \lt s$$ and start dealing with a quantity like $$\sqrt{s}$$ ... is this a sensible assumption ...?So ... assume $$( r + \frac{1}{k} )^2 \lt s$$ ..
then
$$( r + \frac{1}{k} )^2 \lt s$$
$$\Longrightarrow r^2 + \frac{2r}{k} + \frac{1}{k^2} \lt s$$
$$\Longrightarrow r^2 + \frac{1}{k^2} \lt s$$ ... ... since $$\frac{2r}{k} \gt 0$$ ... (but ... how do I justify this step?)
$$\Longrightarrow k^2 \gt \frac{1}{ s - r^2 }$$
But where do we go from here ... seems intuitively that such a $$k \in \mathbb{N}$$ exists ... but how do we prove it ...
(Note that I am assuming that for $$k \in \mathbb{N}$$ that if we show that $$k^2$$ exists, then we know that $$k$$ exists ... is that correct?Hope that someone can clarify the above ...
Help will be much appreciated ...
Peter
===========================================================================================
***NOTE***
In Exercises 1.5.6 to 1.5.8 Bloch gives a series of relations/formulas that may be useful in proving Exercise 1.5.9 (indeed, 1.5.9 (1) and (2) may be useful as well) ... so I am providing Exercises 1.5.6 to 1.5.8 as follows: (for 1.5.9 (1) and (2) please see above)
https://www.physicsforums.com/attachments/7024
View attachment 7025
I am currently focused on Section 1.5: Constructing the Rational Numbers ...
I need help with Exercise 1.5.9 (3) ...Exercise 1.5.9 reads as follows:
View attachment 7023We are at the point in Bloch's book where he has just defined/constructed the rational numbers, having previously defined/constructed the natural numbers and the integers ... so (I imagine) at this point we cannot assume the existence of the real numbers.
Basically Bloch has defined/constructed the rational numbers as a set of equivalence classes on $$\mathbb{Z} \times \mathbb{Z}^*$$ and then has proved the usual fundamental algebraic properties of the rationals ...Now ... we wish to prove that for $$r, s \in \mathbb{Q}$$ where $$r \gt 0$$ and $$s \gt 0$$ that:
If $$r^2 \lt s$$ then there is some $$k \in \mathbb{N}$$ such that $$( r + \frac{1}{k} )^2 \lt s$$ ... ...
Solution Strategy
Prove that there exists a $$k \in \mathbb{N}$$ such that $$( r + \frac{1}{k} )^2 \lt s$$ ... BUT ... without in the proof involving real numbers like $$\sqrt{2}$$ because we have only defined/constructed $$\mathbb{N}, \mathbb{Z}$$, and $$\mathbb{Q}$$ ... so I am assuming that we cannot take the square root of the relation $$( r + \frac{1}{k} )^2 \lt s$$ and start dealing with a quantity like $$\sqrt{s}$$ ... is this a sensible assumption ...?So ... assume $$( r + \frac{1}{k} )^2 \lt s$$ ..
then
$$( r + \frac{1}{k} )^2 \lt s$$
$$\Longrightarrow r^2 + \frac{2r}{k} + \frac{1}{k^2} \lt s$$
$$\Longrightarrow r^2 + \frac{1}{k^2} \lt s$$ ... ... since $$\frac{2r}{k} \gt 0$$ ... (but ... how do I justify this step?)
$$\Longrightarrow k^2 \gt \frac{1}{ s - r^2 }$$
But where do we go from here ... seems intuitively that such a $$k \in \mathbb{N}$$ exists ... but how do we prove it ...
(Note that I am assuming that for $$k \in \mathbb{N}$$ that if we show that $$k^2$$ exists, then we know that $$k$$ exists ... is that correct?Hope that someone can clarify the above ...
Help will be much appreciated ...
Peter
===========================================================================================
***NOTE***
In Exercises 1.5.6 to 1.5.8 Bloch gives a series of relations/formulas that may be useful in proving Exercise 1.5.9 (indeed, 1.5.9 (1) and (2) may be useful as well) ... so I am providing Exercises 1.5.6 to 1.5.8 as follows: (for 1.5.9 (1) and (2) please see above)
https://www.physicsforums.com/attachments/7024
View attachment 7025