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

(a) -2; (b) -2; (c) 2.444...; (d) 1; (e) 2; (f) \( \sqrt{2} \).

Step 1: Understanding the Concept (Set a)

The set given is \( S = \{-10, -8, -6, -4, -2\} \). An upper bound is a number greater than or equal to all numbers in the set. For this finite set, the maximum value of the set itself is the least upper bound. Hence, for (a), the least upper bound is \(-2\).

Step 2: Understanding the Concept (Set b)

The set \( S = \{-2, -2.1, -2.11, -2.111, -2.1111, \ldots\} \) displays a decreasing pattern, approaching a particular value. Recognizing that with more terms in the sequence, the terms approach \(-2.1111\ldots\), we see that the least upper bound is \(-2\). This is because all terms are less than or equal to \(-2\), the limit in this context.

Step 3: Understanding the Concept (Set c)

The set \( S = \{2.4, 2.44, 2.444, 2.4444, \ldots\} \) increases with more decimal places filled with 4s. The sequence approaches 2.44444... The least upper bound, \( \sup S \), is exactly 2.4444... which is a repeating decimal equal to \( \frac{22}{9} \).

Step 4: Understanding the Concept (Set d)

For the set \( S = \left\{1 - \frac{1}{2}, 1 - \frac{1}{3}, 1 - \frac{1}{4}, 1 - \frac{1}{5}, \ldots\right\} \), each term is smaller than 1 as \( \frac{1}{n} \) decreases to zero making limits head towards 1. Thus, the least upper bound is 1.

Step 5: Understanding the Concept (Set e)

For the set \( S = \left\{x: x = (-1)^{n} + \frac{1}{n}, n \text{ a positive integer}\right\} \), values oscillate between \(1 + \frac{1}{n}\) and \(-1 + \frac{1}{n}\). The least upper bound occurs when \( n = 1 \), which leads to \(1 + 1 = 2\). Therefore, the least upper bound is 2.

Step 6: Understanding the Concept (Set f)

For \( S = \{x: x^2 < 2, x \text{ a rational number} \}\), the upper bound is rooted in investigating potential values understanding partial inequality \( x^2 < 2 \) leads to maximum \( x \) as \( \sqrt{2} \). Rational numbers cannot exactly express \( \sqrt{2} \) as it is irrational, hence it's least upper bound is \( \sqrt{2} \).

Key Concepts

upper bounds

An upper bound of a set is a concept that arises frequently in calculus, particularly when working with sequences and series. An upper bound refers to a number that is greater than or equal to all the elements in a given set. This concept is vital in understanding the behavior of sets and sequences as it helps identify the limits within which a set operates. For example, if we consider the set \( S = \{1, 2, 3, 4, 5\} \), any number greater than or equal to 5 can serve as an upper bound. In other words, the upper bound is something like a ceiling for the set. Having a clear notion of upper bounds also introduces us to the least upper bound (or supremum), which is the smallest of all possible upper bounds for a set. This concept is important because it allows us to precisely define the limits of a set, especially when discussing the convergence of sequences in calculus.
Upper bounds are crucial in optimization problems where identifying the maximum value that a set can attain is necessary. By understanding upper bounds, students gain deeper insights into the mechanics of limits and continuity in real analysis.

sets in calculus

Sets are fundamental building blocks in calculus, providing a collection of elements or numbers that share specific properties. In calculus, we often deal with sets of real numbers to describe intervals or sequences. Understanding the nature of these sets is essential for solving problems related to limits and continuity.
For example, consider a set defined by an expression or rule like \( S = \{-2, -2.1, -2.11, -2.111, -2.1111, \ldots\} \). This set represents a sequence of numbers that approach a limit. Identifying sets and their properties allows us to analyze functions, solve equations, and explore limits.

• Discrete sets: which contain distinct and separate elements.
• Continuous sets: like intervals that contain a range of elements.

By evaluating these characteristics, you can determine the maximum or minimum values within a set, which ties into the concept of upper and lower bounds. Sets in calculus also play an integral role in integration and differentiation processes, enabling the calculation of areas under curves and rates of change.

real analysis

Real analysis is a branch of mathematics dealing with real numbers and real-valued functions. It plays a pivotal role in understanding concepts such as continuity, limits, derivatives, and integrals. One of the core ideas in real analysis is the study of sequences and their convergence, which ties directly into the notion of least upper bounds. Real analysis provides the rigorous foundation needed to explore these concepts in depth.
It formalizes the intuitive ideas of calculus, ensuring that definitions and theorems are exact and applicable to various scenarios. For instance, the Bolzano-Weierstrass theorem, which is a key result in real analysis, confirms that any bounded sequence has at least one convergent subsequence. This directly supports our understanding of limit points and least upper bounds in sequences. In essence, real analysis helps refine and ensure the accuracy of calculus concepts, providing tools to analyze and interpret mathematical phenomena with precision.

calculus sequence

A calculus sequence refers to an ordered list of numbers that often follows a specific pattern or rule. Sequences are fundamental objects of study in calculus and real analysis because they allow us to investigate convergence and limits, both crucial components of calculus theory. When discussing sequences, terms such as limits, convergence, and bounds frequently come into play.
For example, consider the sequence \( S = \{2.4, 2.44, 2.444, 2.4444, \ldots\} \). This sequence is increasing and approaches a particular value, which can be determined as its least upper bound. It's helpful to analyze such sequences in terms of their limiting behavior, as it allows us to describe their ultimate behavior more rigorously.

• Understanding sequences: reveals insights into the strengths and limitations of functions in calculus.
• The convergence of a sequence: implies that the terms are getting arbitrarily close to a certain number, which is key to solving problems related to continuity and differentiability.

By examining these sequences, students can deepen their understanding of function behavior and the application of calculus concepts.