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Chapter 5: Problem 40

Use the well-ordering property to show that if \(x\) and \(y\) are real numbers with \(x1 /(y-x)\) . Then show that there is a rational number \(r\) with denominator \(A\) between \(x\) and \(y\) by looking at the numbers \(\lfloor x\rfloor+ j / A,\) where \(j\) is a positive integer.]

Short Answer

Expert verified
Given \(x<y\), there exists a rational number \(r\) such that \(x<r<y\) by choosing \(r=\lfloor x \rfloor + \frac{j_0}{A}\) and appropriate \(A\) and \(j_0\).

Step by step solution

01

- Understand the well-ordering property

The well-ordering property states that any non-empty set of non-negative integers contains a least element. This means we can find the smallest element in the set of non-negative integers.
02

- Apply the Archimedean property

The Archimedean property asserts that for any two real numbers, there exists a positive integer that exceeds their difference. In this case, given real numbers \(x\) and \(y\) with \(x < y\), we need to find a positive integer \(A\) such that \(A > \frac{1}{y-x}\).
03

- Find a suitable positive integer \(A\)

Using the Archimedean property, choose a positive integer \(A\) such that \(A > \frac{1}{y-x}\). This ensures that the distance between consecutive fractions with denominator \(A\) is less than \(y - x\).
04

- Consider fractions with denominator \(A\)

Let鈥檚 look at fractions of the form \(\lfloor x \rfloor + \frac{j}{A}\) for integer values of \(j\). These fractions are rational numbers with a common denominator \(A\).
05

- Identify the smallest \(j\) for which \(\lfloor x \rfloor + \frac{j}{A} > x\)

Since \(x\) is a real number, there will be some integer \(j\) such that \(\lfloor x \rfloor + \frac{j}{A}\) is slightly greater than \(x\). Let this smallest such \(j\) be \(j_0\).
06

- Ensure \(\lfloor x \rfloor + \frac{j_0}{A} < y\)

Given the choice of \(A\) from step 3, since \(A > \frac{1}{y-x}\), the gap between consecutive fractions is less than \(y-x\). Therefore, \(\lfloor x \rfloor + \frac{j_0}{A} < y\).
07

- Confirm the rational number \(r\)

Thus, the rational number \(r = \lfloor x \rfloor + \frac{j_0}{A}\) satisfies \(x < r < y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Archimedean Property
The Archimedean Property is a fundamental principle in real analysis. It asserts that for any two positive real numbers, there is always an integer larger than their ratio. In simpler terms, no matter how small a positive real number may be, there is always a positive integer greater than this number. For instance, given two real numbers, say \(x\) and \(y\) with \(x < y\), you can always find an integer \(A\) such that \(A > \frac{1}{y-x}\). This is crucial in proving the density of rational numbers between any two real numbers. By using this property, we ensure that the gap between these rational numbers can be made smaller than any given real interval.
Rational Numbers
Rational numbers are numbers that can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). Rational numbers include both positive and negative numbers, as well as zero. They are crucial in the real number system because they fill the gaps between integers.
An important characteristic of rational numbers is their density property: between any two real numbers, there exists a rational number. This is what we leverage in problems like finding a rational number between two given real numbers \(x\) and \(y\).
Real Numbers
Real numbers include all the numbers on the number line. This set comprises both rational and irrational numbers. Rational numbers are those that can be expressed as fractions, while irrational numbers cannot be written as simple fractions. Examples of irrational numbers include \(\pi\) and \(\sqrt{2}\).
Real numbers are essential as they provide a complete and continuous number line where every point corresponds to a real number. This continuity is vital in calculus and analysis. Understanding the well-ordering property of real numbers helps in comprehending sequences, limits, and the density argument used in placing rational numbers between any two real numbers.
Least Element
The least element in a set is the smallest element according to the order defined on the set. In the context of non-negative integers, the well-ordering property states that any non-empty set of these integers has a least element. This means you can always find the smallest integer within such a set.
When we apply this concept along with the Archimedean property, we can find an integer \(j\) such that the fraction \(\lfloor x \rfloor + \frac{j}{A}\) is the smallest rational number greater than \(x\). This is key to identifying the exact rational number that lies between two real numbers 'x' and 'y'.

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