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Chapter 2: Problem 4

Show that \(|x-a|<\varepsilon\) if and only if \(a-\varepsilon

Short Answer

Expert verified
The two statements \(|x-a|<\varepsilon\) and \(a-\varepsilon<x<a+\varepsilon\) are equivalent because they both represent the same condition in mathematical notation - \(x\) is within a distance less than \(\varepsilon\) from \(a\).

Step by step solution

01

Start from one part of the statement

Consider the statement \(|x-a|<\varepsilon\). The absolute value, \(|x-a|\), represents the positive distance between \(x\) and \(a\). If this distance is less than \(\varepsilon\), then \(x\) must lie within the interval \((a-\varepsilon , a+\varepsilon)\). Thus, we can write, \(|x-a|<\varepsilon\) as \(a-\varepsilon<x<a+\varepsilon\).
02

Proving the converse

Now, suppose that \(a-\varepsilon<x<a+\varepsilon\). This implies that \(x\) lies within the interval \((a-\varepsilon , a+\varepsilon)\). The distance between \(a\) and \(x\) (i.e., \(|x-a|\)) must therefore be less than \(\varepsilon\). Thus \(a-\varepsilon<x<a+\varepsilon\) implies \(|x-a|<\varepsilon\).

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Key Concepts

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

Real Analysis
Real analysis is a branch of mathematics dealing with real numbers and the analytical properties of real-valued sequences and functions. At its core, real analysis involves rigorous considerations of the notions of convergence, continuity, and limits.

Central to real analysis is the understanding of inequalities, which in themselves are statements about the relative size of two values. Absolute value inequalities, such as the one in our exercise, are used to describe the distance of any real number from a fixed point on the number line. This relates directly to the concepts of boundedness and intervals, which are fundamental in real analysis. These intervals can describe domains of functions, ranges, or the proximity of a sequence to its limit.
Epsilon-Delta Definition
The epsilon-delta definition is a formal mathematical expression used to precisely define the limit of a function as it approaches a particular point. It is represented as: For every number \( \varepsilon > 0 \) there exists a number \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \) then \( |f(x) - L| < \varepsilon \) where \( L \) is the limit value of the function as \( x \) approaches \( c \).

This definition can seem abstract, but it essentially describes a way to gauge how closely \( f(x) \) can get to \( L \) by restricting the distance between \( x \) and \( c \) to be within some small number \( \delta \). The epsilon-delta concept hence builds a bridge between intuitive and formal notions of limits, crucial for understanding continuity and convergence within real analysis.
Mathematical Proofs
Mathematical proofs are the cornerstone of mathematics, providing a system by which mathematicians can verify the truth of statements. Through proofs, concepts and theorems are rigorously justified using logical reasoning and previously established results.

Proofs can take many forms, such as direct, contrapositive, or by contradiction. In the case of our absolute value inequality, the proof is direct: we prove the implication in both directions to show the 'if and only if' relationship. Proving statements like 'if \(|x-a|<\varepsilon\) then \(a-\varepsilon

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