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# 数学代写|数学分析作业代写Mathematical Analysis代考|MA426001 Completeness

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## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Completeness

The mathematicians of antiquity had a clear understanding of the existence of irrational numbers, and mathematicians through the ages understood that irrational numbers are gaps inside the rational number field. Thus it was quite well understood that the rational field is not complete. It took some twentyfour centuries for a rigorous definition of the real number field as a complete ordered field to materialize. The definitions and some of the results in this section parallel those in section 1.2. For example, the proof of the Bolzano-Weierstrass property of bounded sets (theorem 1.2.10) includes a proof of the nested interval theorem, which is a very special case of the Cantor intersection theorem. Another highlight of this section is Baire’s theorem, which is one of the cornerstones upon which functional analysis is built. We will establish the completeness of the $l^{p}$ spaces as well as the function space $\mathcal{C}[a, b]$, which will pave the way for a number of interesting applications begun in the section and continued in the section exercises.

Definition. A sequence $\left(x_{n}\right)$ in a metric space $X$ is said to be a Cauchy sequence if, for every $\epsilon>0$, there is a natural number $N$ such that,
$$\text { for all } m, n>N, d\left(x_{n}, x_{m}\right)<\epsilon \text {. }$$

Theorem 4.6.1. A convergent sequence is a Cauchy sequence.
Proof. Let $\lim x_{n}=x$, and let $\epsilon>0$. There exists a natural number $N$ such that, for $n>N, d\left(x_{n}, x\right)<\epsilon / 2 .$ Now, for $m, n>N, d\left(x_{n}, x_{m}\right) \leq d\left(x_{n}, x\right)+d\left(x, x_{m}\right)<\epsilon .$

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Compactness

A clear manifestation of sequential compactness can be seen in examples 7 and 8 in section 1.2, where we proved the boundedness of continuous functions and their uniform continuity on a compact interval. We urge the reader to re-examine these two examples. This section opens with the topological (non-sequential) definition of compactness and the establishment of the general characteristics of compact spaces. This is done in order to avoid the duplication of definitions and results in the corresponding section in chapter 5. The various equivalent characterizations of compact metric spaces are discussed, and then we prove two famous theorems: Tychonoff’s theorem and the Heine-Borel theorem. The section concludes with an illuminating application on closed convex sunsets of $\mathbb{R}^{n}$.

Definition. A metric space $X$ is said to be compact if every open cover of $X$ contains a finite subcover of $X$. The definitions of open covers and subcovers have been stated in section $4.5$.

Example 1. The collection $\mathcal{U}={(-n, n): n \in \mathbb{N}}$ of open subsets of $\mathbb{R}$ is an open cover of $\mathbb{R}$ that contains no finite subcover. Therefore $\mathbb{R}$ is not compact.

Example 2. The sequence of open intervals $\mathcal{U}=\left{I_{n}=(1 / n, 1-1 / n): n \geq 3\right}$ covers the open interval $(0,1)$, but no finite subset of $\mathcal{U}$ covers $(0,1)$. Thus $(0,1)$ is not compact.

Definition. Let $K$ be a subset of a metric space $X$. We say that $K$ is a compact subset (or a compact subspace) of $X$ if it is compact in the restricted metric.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Completeness

$$\text { for all } m, n>N, d\left(x_{n}, x_{m}\right)<\epsilon \text {. }$$ 定理 4.6.1。收敛序列是柯西序列。 证明。让 $\lim x_{n}=x$ ，然后让 $\epsilon>0$. 存在一个自然数 $N$ 这样，对于 $n>N, d\left(x_{n}, x\right)<\epsilon / 2$.现在，对于 $m, n>N, d\left(x_{n}, x_{m}\right) \leq d\left(x_{n}, x\right)+d\left(x, x_{m}\right)<\epsilon$

## 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。