Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(S=\left\\{x: x\right.\) is rational and \(\left.x^{2}<2\right\\} .\) Convince yourself that \(S\) does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence of rational numbers \(1,1.4,1.41,1.414, \ldots\), has no limit within the rational numbers.

Step-by-Step Solution

Answer

The sequence has no least upper bound among rationals but \( \sqrt{2} \) is the least upper bound in the real numbers.

Step 1: Understand the Set S

The set \( S \) consists of all rational numbers \( x \) such that \( x^2 < 2 \). This means we are considering all rational numbers whose squares are less than 2.

Step 2: Determine Potential Upper Bound in Rationals

To find an upper bound in the rational numbers, consider a number \( x \) such that \( x^2 = 2 \). The number \( \sqrt{2} \) satisfies this equation, but it is irrational. We need to check if there exists a rational number that can be a least upper bound for \( S \).

Step 3: Evaluate if Least Upper Bound Exists in Rationals

Suppose there exists a least upper bound, \( x_0 \), which is a rational number. Then \( x_0^2 < 2 \), which implies there is a rational number \( x_0 + \epsilon \) (for some small positive \( \epsilon \)) such that \((x_0 + \epsilon)^2 < 2\). Thus, \( x_0 \) cannot be the least upper bound, as exceeding it by a rational still lies in \( S \). This shows \( S \) has no least upper bound in the rationals.

Step 4: Examine Bounds in Real Numbers

Now consider the real numbers. The number \( \sqrt{2} \) is indeed an upper bound, as for any rational number \( x < \sqrt{2} \), \( x^2 < 2 \). Furthermore, there are no smaller numbers than \( \sqrt{2} \) that are also the complete least upper bound, since any real number less than \( \sqrt{2} \) does not satisfy \( x^2 \geq 2 \). Thus, \( \sqrt{2} \) is the least upper bound in the real numbers.

Key Concepts

Rational Numbers

Rational numbers are the numbers that can be expressed as the quotient or fraction of two integers. They have the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). This means every rational number has a finite or repeating decimal representation.

Some key characteristics of rational numbers include:

• Every integer is a rational number because it can be expressed as a fraction with denominators of 1.
• Rational numbers include fractions like \( \frac{1}{2} \), \( \frac{2}{3} \), and even negative fractions.
• Rational numbers are dense, which means between any two rational numbers, there is another rational number.

In the context of our problem, we are analyzing whether the set of rational numbers whose squares are less than 2 has a least upper bound. However, since rational numbers cannot provide an exact square root of 2, a perfect least upper bound like \( \sqrt{2} \) isn't rational. This difference makes it challenging for rational numbers to serve as complete upper limits in certain sets, as is digitized in the example of our exercise.

Irrational Numbers

Irrational numbers are those that cannot be represented as a simple fraction of two integers. Their decimal expansion is non-terminating and non-repeating. Common examples of irrational numbers include \( \pi \) and \( \sqrt{2} \).

Let's delve into some important features of irrational numbers:

• Irrational numbers fill the gaps between rational numbers on the real number line.
• Their decimal expansions go on forever without forming a repeating pattern, which is what distinguishes them from rationals.
• Irrational numbers are useful for capturing precise values where rational numbers fall short.

In the exercise, we find that \( \sqrt{2} \) is an irrational number. It's crucial in demonstrating that while \( S \) lacks a least upper bound among rational numbers, \( \sqrt{2} \) provides such a bound within the real numbers.Distance between a rational and its irrational limit can be infinitely reduced, yet never actually met, defining the essence of irrationality.

Real Analysis

Real Analysis is a branch of mathematics that deals with the real numbers and the real-valued sequences and functions. It provides the formal underpinning for calculus and involves understanding limits, continuity, differentiability, and the behavior of sequences and series.

Real Analysis is important for several reasons:

• It forms the theoretical basis for many calculus concepts like limits and integrals.
• It investigates the properties of real-valued functions and the completeness of the real numbers.
• Least Upper Bound (LUB) or supremum is a critical part of understanding boundedness, a concept studied in Real Analysis.

In the scenario provided by our exercise, Real Analysis aids in understanding why \( \sqrt{2} \) exists as a least upper bound in the set of real numbers for \( S \), even though such a boundary doesn't exist within the rationals. The distinction made between the behavior of sequences in rational and real numbers exemplifies the precision Real Analysis aims to achieve.