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Chapter 2: Problem 5

If \(a

Short Answer

Expert verified
The inequality \(|x-y|<b-a\) can be proven by understanding that within an open interval, the distance between any two points (in this case \(x\) and \(y\)) will be less than the distance spanning the entire interval. Geometrically, this suggests that no matter where \(x\) and \(y\) are in the interval \((a,b)\), the distance between them will always be less than the length of the full interval.

Step by step solution

01

Understand the given inequalities

First, consider the inequalities \(a<x<b\) and \(a<y<b\). These inequalities mean that both \(x\) and \(y\) fall within the open interval \((a, b)\) on the real number line.
02

Define the absolute value

Next, consider the meaning of \(|x-y|\). This quantity represents the absolute value of the difference of \(x\) and \(y\), which is interpreted as the distance between \(x\) and \(y\) on the number line.
03

Examine the relation between distance and interval

By the definition of an open interval, the distance between any two points inside the open interval \((a,b)\) would be less than the total distance of the interval, which is \(b-a\). Therefore, we get \(|x-y|<b-a\).
04

Geometric interpretation

Geometrically, this means that no matter where \(x\) and \(y\) are located in the interval \((a,b)\), the distance between them will always be smaller than the length of the interval itself.

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Most popular questions from this chapter

Let \(X\) and \(Y\) be nonempty sets and let \(h: X \times Y \rightarrow \mathbb{R}\) have bounded range in \(\mathbb{R}\). Let \(F: X \rightarrow \mathbb{R}\) and \(G: Y \rightarrow \mathbb{R}\) be defined by $$ F(x):=\sup \\{h(x, y): y \in Y\\}, \quad G(y):=\sup \\{h(x, y): x \in X\\} $$ Establish the Principle of the Iterated Suprema: $$ \sup \\{h(x, y): x \in X, y \in Y\\}=\sup \\{F(x): x \in X\\}=\sup \\{G(y): y \in Y\\} $$ We sometimes express this in symbols by $$ \sup _{x, y} h(x, y)=\sup _{x} \sup _{y} h(x, y)=\sup _{y} \sup _{x} h(x, y) . $$

Let \(I_{n}:=[0,1 / n]\) for \(n \in \mathbb{N}\). Prove that \(\bigcap_{n=1}^{\infty} I_{n}=\\{0\\}\).

(a) If \(0

(a) Give the first four digits in the binary representation of \(\frac{1}{3}\). (b) Give the complete binary representation of \(\frac{1}{3}\).

Let \(S_{4}:=\left\\{1-(-1)^{n} / n: n \in \mathbb{N}\right\\}\). Find inf \(S_{4}\) and sup \(S_{4}\).
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