Problem 82 - Calculus Early Transcendentals: Pearson New International Edition

Question

A number \(b\) is called an upper bound for a set \(S\) of numbers if \(x \leq b\) for all \(x\) in \(S\). For example \(5,6.5\), and 13 are upper bounds for the set \(S=\\{1,2,3,4,5\\}\). The number 5 is the least upper bound for \(S\) (the smallest of all upper bounds). Similarly, 1.6, 2, and \(2.5\) are upper bounds for the infinite set \(T=\\{1.4,1.49,1.499,1.4999, \ldots\\}\), whereas \(1.5\) is its least upper bound. Find the least upper bound of each of the following sets. (a) \(S=\\{-10,-8,-6,-4,-2\\}\) (b) \(S=\\{-2,-2.1,-2.11,-2.111,-2.1111, \ldots\\}\) (c) \(S=\\{2.4,2.44,2.444,2.4444, \ldots\\}\) (d) \(S=\left\\{1-\frac{1}{2}, 1-\frac{1}{3}, 1-\frac{1}{4}, 1-\frac{1}{5}, \ldots\right\\}\) (e) \(S=\left\\{x: x=(-1)^{n}+1 / n, n\right.\) a positive integer \(\\}\); that is, \(S\) is the set of all numbers \(x\) that have the form \(x=(-1)^{n}+1 / n\), where \(n\) is a positive integer. (f) \(S=\left\\{x: x^{2}<2, x\right.\) a rational number \(\\}\)

Step-by-Step Solution

Answer

Set a: -2, Set b: -2, Set c: 2.4̅, Set d: 1, Set e: 1, Set f: 1.414...

Step 1: Analyze Set (a)

The set given is \( S = \{-10, -8, -6, -4, -2\} \). We are tasked to find the least upper bound (LUB) of this set. All elements of this set are distinct integers and are finite. By examining the set, the greatest number in the set is -2. Thus, -2 is the least upper bound for this set since any number greater than -2 would also act as an upper bound.

Step 2: Analyze Set (b)

The set is \( S = \{-2, -2.1, -2.11, -2.111, -2.1111, \ldots\} \). This sequence gets increasingly negative towards -2.1111... indefinitely. The pattern suggests it's approaching -2.111\overline{1}, but each term is still less than -2. Hence, the least upper bound is -2.

Step 3: Analyze Set (c)

Set \( S = \{2.4, 2.44, 2.444, 2.4444, \ldots\} \) is a sequence that increases towards 2.4444... The terms are getting arbitrarily close to \(2.\overline{4}\), a repeating decimal. Thus, the least upper bound of this set is 2.444... or \(2.\overline{4}\).

Step 4: Analyze Set (d)

The set is given as \( S = \left\{1-\frac{1}{2}, 1-\frac{1}{3}, 1-\frac{1}{4}, 1-\frac{1}{5}, \ldots\right\} \). This represents the sequence 1 minus positive fractions that get smaller, hence approaching 1. Since there is no term greater than 1 but terms are approaching it, 1 is the least upper bound.

Step 5: Analyze Set (e)

Given set \( S = \left\{x: x=(-1)^{n}+\frac{1}{n}\right\} \). When \(n\) is even, \(x = 1 + \frac{1}{n}\); when \(n\) is odd, \(x = -1 + \frac{1}{n}\). The sequence for even \(n\) approaches 1, and for odd \(n\), it approaches -1. The greatest value will always be slightly above 1 when \(n\) is even, hence the least upper bound is slightly above 1, yet approaches exactly 1 as \(n\) increases indefinitely. Thus, the least upper bound is 1.

Step 6: Analyze Set (f)

Set \( S = \left\{x: x^{2}<2, \right.\) a rational number \( \right\}\). This set includes all rational numbers \(x\) such that \(x^{2} \) is less than 2. The rational number closest to \(\sqrt{2}\) is approximately 1.414... However, since \(\sqrt{2}\) is irrational, we approximate rational numbers below it. LUB will be the largest rational number defined by sequences converging to this irrational boundary, approximately \(1.414...\)

Key Concepts

Upper Bound

An upper bound of a set is a number that is greater than or equal to every element in that set. Let's take a closer look at this concept using some examples.
Consider the set of numbers: \( S = \{1, 2, 3, 4, 5\} \). Numbers such as 5, 6.5, and even 13 can be upper bounds because they are all greater than any number in \( S \).
The smallest of these, 5, is called the least upper bound (or supremum), which is the key point of focus when dealing with upper bounds.Finding the least upper bound involves considering the smallest possible number that still acts as an upper bound. It’s an important concept especially in problems involving infinite sets, where not every upper bound is obvious.
For example, consider \( T = \{1.4, 1.49, 1.499, 1.4999, \ldots \} \). The least upper bound here is 1.5, which the sequence converges towards without actually reaching beyond it.

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and real-valued functions. It is primarily focused on limits, series, and the rigorous study of calculus.
In real analysis, the concept of bounds plays a pivotal role, particularly in understanding the behavior of sequences and their limits.The greatest insight offered by real analysis is perhaps its approach to completeness, where every bounded set of real numbers has a least upper bound. This is a key property distinguishing real numbers from rationals.

• Every Cauchy sequence in real analysis converges to a limit that's within the reals.
• Consider the set of numbers that approach a value but do not exceed it, such as sequences converging to \( \sqrt{2} \).

Real analysis uses such structures to reason about continuity, differentiation, and functional approximations, making it central to advanced calculus studies.

Sets and Sequences

Sets and sequences are foundational concepts in mathematics, especially when studying limits and series.A **set** is a collection of distinct objects, which can be made up of numbers, vectors, or other elements. Sets can be finite, like \( S = \{-10, -8, -6, -4, -2\} \) in our exercise, or infinite, like the sequence \( S = \{2.4, 2.44, 2.444, \ldots\} \).A **sequence** is essentially an ordered list of elements, typically numbers, often showing some pattern or rule. When we describe sequences like \( \{1-rac{1}{2}, 1-rac{1}{3}, \ldots \} \), we're looking at terms that get inexorably closer to a certain value.Study of sequences often involves limits, as sequences approach ('converge to') a specific number, such as in the aforementioned cases. If we can find a number that a sequence converges to without exceeding, that becomes our least upper bound.
Sequences and their convergence are fundamental in calculus and analysis, offering insights not just into limits, but also into phenomena like continuity and differentiability.