## 数学代写|实分析作业代写Real analysis代考|MAST20026

2022年7月26日

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## 数学代写|实分析作业代写Real analysis代考|Infinite Limits

If a monotone increasing sequence $\left{a_{n}\right}$ is bounded above then by Theorem $3.3 .2$ the sequence converges. If the sequence $\left{a_{n}\right}$ is not bounded above, then for each positive real number $M$ there exists $n_{b} \in \mathbb{N}$ such that $a_{n} \geq M$ for all $n \geq n_{n}$. Since the real number $M$ can be taken to be arbitrarily large, this is usually expressed by saying that the sequence $\left{a_{n}\right}$ diverges to $\infty$. We make this concept precise, not only for monotone sequences, but for any sequence of real numbers with the following definition.

DEFINITION 3.3.6 Let $\left{a_{n}\right}$ be a sequence of real numbers. We say that $\left{a_{n}\right}$ approaches infinity, or that $\left{a_{n}\right}$ diverges to $\infty$, denoted $a_{n} \rightarrow \infty$, if for every positive real number $M$, there exists an integer $n_{o} \in \mathbb{N}$ such that
$$a_{n}>M \quad \text { for all } n \geq n_{o}$$
We will also use the notation $\lim {n \rightarrow \infty} a{n}=\infty$ to denote that $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$. The concept of $a_{n} \rightarrow-\infty$ is defined similarly.

THEOREM 3.3.7 If $\left{a_{n}\right}$ is monotone increasing and not bounded above, then $a_{n} \rightarrow \infty$ as $n \rightarrow \infty$

As a consequence of Theorems $3.3 .2$ and 3.3.7, every monotone increasing sequence $\left{a_{n}\right}$ either converges to a real number (if the sequence is bounded above) or diverges to $\infty$. In either case,
$$\lim {n \rightarrow \infty} a{n}=\sup \left{a_{n}: n \in \mathbb{N}\right}$$

## 数学代写|实分析作业代写Real analysis代考|Subsequences and the Bolzano-Weierstrass Theorem

In this section, we will consider subsequences and subsequential limits of a given sequence of real numbers. One of the key results of the section is that every bounded sequence of real numbers has a convergent subsequence. This result, also known as the sequential version of the Bolzano-Weierstrass theorem, is one of the fundamental results of real analysis.

DEFINITION 3.4.1 Let $(X, d)$ be a metric space. Given a sequence $\left{p_{n}\right}$ in $X$, consider a sequence $\left{n_{k}\right}_{k=1}^{\infty}$ of positive integers such that $n_{1}<n_{2}<$ $n_{3}<\ldots$ Then the sequence $\left{p_{n_{k}}\right}_{k=1}^{\infty}$ is called a subsequence of the sequence $\left{p_{n}\right}$

If the sequence $\left{p_{n_{k}}\right}$ converges, its limit is called a subsequential limit of the sequence $\left{p_{n}\right}$. Specifically, a point $p \in X$ is a subsequential limit of the sequence $\left{p_{n}\right}$ if there exists a subsequence $\left{p_{n_{k}}\right}$ of $\left{p_{n}\right}$ that converges to $p$. Also, given a sequence $\left{p_{n}\right}$ in $\mathbb{R}$, we say that $\infty$ is a subsequential limit of $\left{p_{n}\right}$ if there exists a subsequence $\left{p_{n_{k}}\right}$ so that $p_{n_{k}} \rightarrow \infty$ as $k \rightarrow \infty$. Similarly for $-\infty$.

EXAMPLES 3.4.2 (a) Consider the sequence $\left{1-(-1)^{n}\right}$. If $n$ is even, then $a_{n}=0$, and if $n$ is odd, then $a_{n}=2$. Thus 0 and 2 are subsequential limits of the given sequence. That these are the only two subsequential limits are left to the exercises (Exercise 1).

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Infinite Limits

$n \geq n_{n}$. 由于实数 $M$ 可以取任意大，这通常表示为序列 Veft{a_{n}〉right} 发散到 $\infty$. 我们使 这个概念变得精确，不仅适用于单调序列，而且适用于具有以下定义的任何实数序列。

$$a_{n}>M \quad \text { for all } n \geq n_{o}$$

\im ${\mathrm{n} \backslash$ rightarrow \infty $}$ a ${\mathrm{n}}=\backslash$ sup $\backslash$ eft $\left{a_{-}{n}: \mathrm{n} \backslash\right.$ in \mathbb ${\mathrm{N}}$ right $}$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。