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Chapter 2: Problem 20

(a) If \(0

Short Answer

Expert verified
The inequality \(0<c^{2}<c<1\) is proven for \(0<c<1\) and the inequality \(1<c<c^{2}\) is proven for \(1<c\).

Step by step solution

01

Prove \(0

Given that \(0<c<1\), which means \(c\) is a positive fraction. Squaring any fraction that is between 0 and 1 will result in a smaller number. Thus, the given expression \(0<c^{2}<c<1\) is verified because the square of a fraction \(c\) where \(0<c<1\) will be less than \(c\) and still be greater than 0 and less than 1.
02

Prove \(1

Given that \(1<c\), which means \(c\) is a number greater than 1. Squaring any number greater than 1 gives an even larger number. Thus, the given expression \(1<c<c^{2}\) is verified because the square of a number (\(c^{2}\)) where \(1<c\) will be greater than \(c\) while the number \(c\) is greater than 1.

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical statements that relate expressions, showing that one quantity is less than, greater than, or equal to another. They are foundational in real analysis and allow us to compare values and determine ranges. When working with real numbers, understanding inequalities helps us to solve problems involving limits, continuity, and optimization.

For instance, the exercise provided demonstrates an inequality involving a variable, c. Analyzing inequalities requires a fundamental understanding of number properties. When a number between 0 and 1 is squared, the result is smaller than the original, and when a number greater than 1 is squared, the result is larger. Recognizing such patterns is critical to mastering inequalities and effectively applying them in complex problems in real analysis.
Exponential Relations in Inequalities
Exponents play a significant role when working with inequalities. When an exponent is applied to a base number, it tells us how many times to multiply that number by itself. This operation can significantly affect the value of the base, especially in the context of inequalities.

For example, if we have a number c such that 0
Constructing Mathematical Proofs
Mathematical proofs are logical arguments that establish the truth of a mathematical statement beyond all doubt, using a series of logically connected steps. The proofs for the given exercise demonstrating properties of exponents within inequalities exemplify this process. Proving statements requires not only a thorough understanding of mathematical concepts but also the ability to apply logic and reasoning skills.

In proving the given inequalities, we first accept the premise that a given c falls within a specified range. We then square c and apply known properties of numbers to deduce the new inequalities. This step-by-step approach is mirrored in all proofs: starting from known truths, applying consistent logic, and arriving at a new understanding. Mastering the art of proving assists with deeper comprehension of real analysis, as well as with verifying theorems and solving complex problems with certainty.

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Most popular questions from this chapter

Modify the argument in Theorem \(2.4 .7\) to show that there exists a positive real number \(u\) such that \(u^{3}=2\).

Let \(S \subseteq \mathbb{R}\) be nonempty. Show that \(u \in \mathbb{R}\) is an upper bound of \(S\) if and only if the conditions \(t \in \mathbb{R}\) and \(t>u\) imply that \(t \notin S\).

Let \(a, b, c, d\) be numbers satisfying \(0

Let \(X\) be a nonempty set, and let \(f\) and \(g\) be defined on \(X\) and have bounded ranges in \(\mathbb{R}\). Show that $$ \sup \\{f(x)+g(x): x \in X\\} \leq \sup (f(x): x \in X\\}+\sup \\{g(x): x \in X\\} $$ and that $$ \inf \\{f(x): x \in X\\}+\inf \\{g(x): x \in X\\} \leq \inf \\{f(x)+g(x): x \in X\\} $$ Give examples to show that each of these inequalities can be either equalities or strict inequalities.

Find all \(x \in \mathbb{R}\) that satisfy the following inequalities. (a) \(|x-1|>|x+1|\), (b) \(|x|+|x+1|<2\).
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