/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Let \(\left\\{f_{n}\right\\}\) b... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 11

Let \(\left\\{f_{n}\right\\}\) be a sequence of real functions. Show that the set \(E\) of points of convergence of the sequence can be written in the form $$ E=\bigcap_{k=1}^{\infty} \bigcup_{N=1}^{\infty} \bigcap_{n=N}^{\infty} \bigcap_{m=N}^{\infty}\left\\{x:\left|f_{n}(x)-f_{m}(x)\right| \leq \frac{1}{k}\right\\} $$

Short Answer

Expert verified
The convergence set \( E \) is defined by ensuring \( \{f_n(x)\} \) is a Cauchy sequence over all \( k \), which follows the nested limit structure in intersections and unions.

Step by step solution

01

Understand the Set of Convergence

The set \( E \) of points of convergence of the sequence \( \{f_n\} \) consists of all points \( x \) where the sequence \( \{f_n(x)\} \) converges to some limit as \( n \rightarrow \infty \). This means we must find \( x \) where the sequence is Cauchy (eventually stabilize as \( n \) increases).
02

Define Cauchy Condition for Uniformly Cauchy Sequence

A sequence \( \{f_n(x)\} \) at \( x \) is Cauchy if for every \( \epsilon > 0 \), there exists a number \( N \) such that for all \( n, m > N \), \( |f_n(x) - f_m(x)| < \epsilon \). This ensures the sequence's convergence.
03

Convert to Countable Forms for \( E \)

Replace \( \epsilon \) with \( \frac{1}{k} \), for \( k = 1, 2, 3, \ldots \). The original arbitrary \( \epsilon \) is represented by the countable family \( \left\{\frac{1}{k}\right\} \). Thus, requiring the inequality \( |f_n(x) - f_m(x)| < \frac{1}{k} \).
04

Build Intersection for Stabilizing Sequence

The condition \( |f_n(x) - f_m(x)| < \frac{1}{k} \) for all \( n, m \) larger than some \( N \) is captured using: \( \bigcap_{n=N}^{\infty} \bigcap_{m=N}^{\infty}\{x:|f_n(x)-f_m(x)| \leq \frac{1}{k}\} \), ensuring all pairs \( n, m \) after \( N \) meet the requirement.
05

Combine \( N \) Requirements

For every specific \( k \), the points where each sequence \( f_n(x) \) begins to stabilize, independent of \( N \), are captured by the union \( \bigcup_{N=1}^{\infty} \). It accommodates any starting point \( N \) fulfilling the Cauchy criterion \( |f_n(x) - f_m(x)| < \frac{1}{k} \).
06

Final Intersection Over All \( k \)

The final intersection \( \bigcap_{k=1}^{\infty} \) ensures convergence for every \( \frac{1}{k} \), hence for every positive \( \epsilon \). It ensures the set \( E \) captures exactly the points \( x \) of convergence.

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Key Concepts

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

Sequence Convergence
In real analysis, sequence convergence is a fundamental concept that helps us understand how sequences behave as they approach a specific value. A sequence \( \{f_n(x)\} \) of real functions is said to converge at a point \( x \) if, as \( n \rightarrow \infty \), the sequence approaches some limit at that point. Specifically, this means that for every positive number \( \epsilon \), no matter how small, there exists a stage in the sequence after which all subsequent terms are within \( \epsilon \) of the limit.This idea is crucial for determining whether a function behaves predictably as you count higher and higher in the sequence. Reliability of behavior over an infinite sequence provides numerous applications such as in calculations, system modeling, and understanding patterns in natural phenomena.
Cauchy Condition
The Cauchy condition is an essential criterion for determining convergence within a sequence. A sequence \( \{f_n(x)\} \) is Cauchy at a point \( x \) if the terms of the sequence get arbitrarily close to each other as \( n \) and \( m \) become large. More formally, for every \( \epsilon > 0 \), there exists a number \( N \) such that for all \( n, m > N \), the distance between the terms, \( |f_n(x) - f_m(x)| \), is less than \( \epsilon \).
  • This means the sequence must "bunch up" as it progresses, not spreading out or diverging.
  • The Cauchy condition can be thought of as a prerequisite for convergence, since it ensures terms are close regardless of an actual limit.
In practical uses, fulfilling the Cauchy condition suggestively hints at convergence, especially in complete metric spaces.
Set Theory
Set theory forms the foundation of much of modern mathematics, establishing a language for discussing collections of objects, such as numbers or functions. In this context, the set \( E \) consists of all the points at which the sequence \( \{f_n\} \) converges. It is constructed using
  • The intersection of sets, \( \bigcap \), which identifies points common to all sets in the intersection.
  • The union of sets, \( \bigcup \), which brings together points from each set into a single set.
Using these operations allows for the description of the convergence set as the intersection of conditions across all \( k \) values to ensure convergence points comply with every finer requirement of the Cauchy condition.
Mathematical Proof
Mathematical proofs are the backbone of verifying statements and claims in mathematics. A proof demonstrating the convergence of \( \{f_n\} \) involves showing that all points meeting the Cauchy condition exist within the constructed set \( E \). For this particular problem, the use of proofs involves:
  • Logical deductions that follow from one step to the next, ensuring each transition is valid.
  • Combining rigorous definitions 鈥� like that of Cauchy sequences and set operations 鈥� to establish the desired results.
The proof given transforms abstract theoretical work into systematic logic, ensuring that the set captured truly contains all points of convergence, thus validating the convergence criteria outlined by the Cauchy condition.

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Most popular questions from this chapter

Let \(f_{n}(x)=\sin n x / \sqrt{n}\). Show that \(\lim _{n} f_{n}=0\) but \(\lim _{n} f_{n}^{\prime}(0)=\infty\).

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