- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

## Homework Statement

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:

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## ... ...

## Homework Equations

... and relevant information ...[/B]We 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 ...

## The Attempt at a Solution

*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)

#### Attachments

Last edited: