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Chapter 2: Problem 8

Let \(\left\\{x_{n}\right\\}\) be a sequence of numbers and \(A_{n}=\left(-\infty, x_{n}\right)\). Show that \(x=\lim\) sup \(x_{n}\) and \(A=\lim \sup A_{n}\) are related in the following way: \((-\infty, x) \subseteq A \subseteq(-\infty, x]\). In other words, \(A\) is equal to either \((-\infty, x)\) or to \((-\infty, x]\).

Short Answer

Expert verified
\( A = (-\infty, x) \) or \( A = (-\infty, x] \).

Step by step solution

01

Review Definitions

Recall that \( \lim \sup x_n \) (limit superior) of a sequence \((x_n)\) is the largest limit of any subsequence of \((x_n)\), or equivalently, \( \lim \sup x_n = \lim_{n \to \infty} ( \sup_{k \geq n} x_k ) \). Similarly, \( A = \lim \sup A_n \) is defined such that an element \( y \) is in \( A \) if and only if for every \( n \), there exists \( k \geq n \) with \( y \in A_k \), implying \( y < x_k \).
02

Show \((-\infty, x) \subseteq A\)

For any \( y < x \), we know that \( y < \lim \sup x_n \). This implies there exists some subsequence \( x_{n_k} \) converging to \( x \) such that \( y < x_{n_k} \) for sufficiently large \( k \). Thus, \( y \in A_{n_k} = (-\infty, x_{n_k}) \) for sufficiently large \( k \). Therefore, \( y \in A \), which proves \((-\infty, x) \subseteq A \).
03

Show \(A \subseteq (-\infty, x]\)

Consider \( y \in A \). By definition of \( A = \lim \sup A_n \), for every \( n \), there exists \( k \geq n \) such that \( y < x_k \), so \( y \in A_k = (-\infty, x_k) \). Therefore, \( y \leq \sup_{k \geq n} x_k \) for all \( n \), implying \( y \leq \lim \sup x_n = x \). Thus, \( A \subseteq (-\infty, x] \).
04

Conclude with Possible Forms for A

Given that \((-\infty, x) \subseteq A \subseteq (-\infty, x]\), \( A \) must take one of two forms: either \((-\infty, x)\) or \((-\infty, x]\). This is because the only options for \( A \) consistent with both inclusions are these intervals.

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Key Concepts

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

Limit Superior
In mathematical analysis, the concept of "limit superior" is essential when studying the behavior of sequences. The limit superior of a sequence \( \{x_n\} \) is commonly denoted as \( \lim \sup x_n \). It refers to the greatest accumulation point of the sequence. To put it simply, it's the highest value that the sequence approaches as it progresses indefinitely without necessarily converging entirely to that point.
  • It is calculated as \( \lim_{n \to \infty} ( \sup_{k \geq n} x_k ) \), where \( \sup \) denotes the supremum or least upper bound.
  • The limit superior helps determine whether a sequence is bounded above and in what manner - a crucial tool in sequence analysis.
  • Additionally, if there are multiple subsequential limits, the limit superior ensures consideration of the "largest" one.
This concept is highly significant in understanding how sequences behave over the long term and plays a pivotal role in real analysis, especially when a sequence does not converge in the traditional sense.
Sequences
Sequences in mathematics are ordered lists of numbers, which are the primary objects of study in analysis. They are often the base structure upon which key calculus and analysis concepts are founded. The study of sequences helps us understand how functions behave as inputs grow indefinitely.
  • A sequence \( \{x_n\} \) is an ordered set of elements where each element is indexed by \( n \), typically a positive integer.
  • Sequences can be finite or infinite, and they can converge to a limit or diverge.
  • Analysts study sequences to explore properties like boundedness, monotonicity, and convergence, which inform on the limit behaviors and stabilization of functions.
By understanding sequences, you can analyze complex functions more effectively and identify any patterns or trends in numerical behavior.
Subsequences
A subsequence is derived from a sequence by selecting some of its elements without changing their original order. Importantly, subsequences maintain the logical order and properties of the original sequence, allowing an in-depth study of its behavior.
  • Suppose you have a sequence \( \{x_n\} \). A subsequence of this sequence is represented as \( \{x_{n_k}\} \).
  • The indices \( n_k \) must strictly increase as \( k \) increases but can skip some indices. For example, you could possibly choose only the elements indexed by even numbers in an original sequence.
  • Analyzing subsequences is crucial because even if the sequence itself does not converge, subsequences may converge to different limits, shedding light on the behavior of the main sequence.
Subsequence analysis is a core component of understanding concepts such as limit superior and is essential in showing complex relationships within sequences.
Set Theory
Set theory is the branch of mathematical logic that deals with sets, or collections of objects. It's a fundamental concept that underlies many areas of mathematics, including analysis, where it provides a rigorous framework for dealing with collections of numbers or points.
  • Sets are typically denoted with curly braces, like \( \{a, b, c\} \), and are central to understanding how sequences behave through operations like unions, intersections, and complements.
  • In the context of the exercise, sets like \((-\infty, x)\) represent the collection of all numbers "less than \( x \)", illustrating a type of interval.
  • Set theory aids in articulating and proving properties and theorems about sequences, such as how their limit behaviors (like limit superior) can be characterized using set operations.
Understanding set theory provides a crucial toolkit for articulating abstract mathematical concepts and is integral to the logical structure needed for formal analysis.

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Most popular questions from this chapter

Let \(\xi=\left(\xi_{1}, \ldots, \xi_{n}\right)\) and \(E \| \xi l^{n}<\infty\), where \(\|\xi\|=+\sqrt{\sum \xi_{i}^{2}}\). Show that $$ \varphi_{\xi}(t)=\sum_{k=0}^{n} \frac{i^{k}}{k !} \mathrm{E}(t, \xi)^{2}+\varepsilon_{n}(t)\|t\|^{n} $$ where \(t=\left(t_{1}, \ldots, t_{n}\right)\) and \(\varepsilon_{n}(t) \rightarrow 0, t \rightarrow 0\).

Using Fatou's lemma, show that $$ \mathrm{P}\left(\underline{\lim } A_{n}\right) \leq \varliminf \mathrm{P}\left(A_{n}\right), \quad \mathrm{P}\left(\overline{\lim } A_{n}\right) \geq \varlimsup \mathrm{lim} \mathrm{P}\left(A_{n}\right) $$

Let \(\xi_{1}, \xi_{2} \ldots\) be a sequence of orthogonal random variables and \(S_{n}=\xi_{1}+\cdots+\xi_{n}\). Show that if \(\sum_{==1}^{\infty} \mathrm{E} \xi_{n}^{2}<\infty\) there is a random variable \(S\) with \(E S^{2}<\infty\) such that Li.m. \(S_{n}=S\), i.e. \(\| S_{n}-\left.S\right|^{2}=E\left|S_{n}-S\right|^{2} \rightarrow 0, n \rightarrow \infty\).

Let \(\left(\xi_{1}, \ldots, \xi_{n}\right)\) be a family of orthogonal random variables. Show that they have the Pythagorean property, $$ \left\|\sum_{i=1}^{n} \xi_{i}\right\|^{2}=\sum_{i=1}^{n}\left\|\xi_{i}\right\|^{2}. $$

Let \(\eta_{1}, \eta_{2}, \ldots\) be an orthonormal system and \(\mathscr{L}=\overline{\mathcal{L}}\left\\{\eta_{1}, \eta_{2}, \ldots\right\\}\) the closed linear manifold spanned by \(\eta_{1}, \eta_{2}, \ldots\) Show that the system is a basis for the (Hilbert) space \(\overline{\mathscr{L}} .\)
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