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Chapter 3: Problem 13

(a) Show that the family \(\mathscr{F}\) of all intervals of the form \(I_{n}=\\{x: 1 /(n+2)

Short Answer

Expert verified
The family \(\mathscr{F}\) covers \(J\) because the intervals \(I_n\) become arbitrarily small and every \(x\) in \(J\) will be covered by some interval. However, no finite subset of \(\mathscr{F}\) can cover \(J\) since there will always be an interval \(I_{N+1}\) not covered by a maximal element \(I_N\) in the finite subset.

Step by step solution

01

Understanding the Infinite Family \(\mathscr{F}\)

The family \(\mathscr{F}\) consists of open intervals \(I_n\) of the form \(1/(n+2)<x<1/n\) for \(n \in \mathbb{N}\). These intervals are designed to get smaller and approach zero as \(n\) increases.
02

Verifying that \(\mathscr{F}\) Covers \(J\)

To prove the family of intervals covers the interval \(J\), we need to show that for every point \(x\) in \(J\), there exists some interval \(I_n\) in \(\mathscr{F}\) such that \(x\) lies within that interval. Since \(1/(n+2)\) becomes smaller and \(1/n\) becomes larger as \(n\) increases, all points in \(J\) with \(0 < x < 1/2\) will be within some interval \(I_n\) for sufficiently large \(n\).
03

Demonstrating no Finite Subfamily of \(\mathscr{F}\) Covers \(J\)

Assume a finite subset of \(\mathscr{F}\), say \(\mathscr{G} = \{I_{n_1}, I_{n_2}, ..., I_{n_k}\}\), covers \(J\). Let \(N = \max\{n_1, n_2, ..., n_k\}\). The interval \(I_{N+1}\) is not in \(\mathscr{G}\), and it contains the point \(1/(N+3)\), which is not covered by any of the intervals in \(\mathscr{G}\), because \(1/(N+3)\) is smaller than the lower bound of any \(I_{n_i}\) in \(\mathscr{G}\). Therefore, no finite subfamily can cover \(J\).

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Key Concepts

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

Infinite Family of Intervals
An infinite family of intervals, such as the one presented in our exercise \(\mathscr{F}\)\, is a set that contains an unlimited number of intervals. These intervals can vary in size, location, and other properties, but they are bound by a rule or formula. In our case, each interval \(I_{n}\) is given by the condition \(1 /(n+2)
Open Intervals in Analysis
In real analysis, an interval is a set of numbers that lies between two endpoints. An open interval does not include its endpoints, which is denoted mathematically as \(a, b\) where \(a\) is the lower bound and \(b\) is the upper bound. In our exercise, the intervals of \(\mathscr{F}\) are all open intervals because they take the form \(1/(n+2)Why use open intervals?Open intervals are significant in analysis for many reasons. One key reason is that they provide a way to discuss continuity and limit processes without the restriction of needing to define function values at the endpoints. Moreover, open intervals can represent an infinite process, as they suggest a sequence of numbers getting perpetually closer to an endpoint without reaching it鈥攁 concept that is pivotal in calculus and limit theory.
Covering Intervals Property
The covering intervals property is an essential concept in real analysis, which refers to the ability of a collection of intervals to completely contain another set. In our exercise, the aim is to show that the infinite family \(\mathscr{F}\) covers the interval \((J = \{x: 0

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Most popular questions from this chapter

\(x_{n}=(\sin (n \pi / 3))(1-(1 / n))\)

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