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

In this section, we introduce the concept of a compact set in the setting of metric spaces. A characterization of the compact subsets of $\mathbb{R}$ is provided in Section 2.4. The notion of a compact set is very important in the study of analysis, and many significant results in the text will depend on the fact that every closed and bounded interval in $\mathbb{R}$ is compact. The modern definition of a compact set given in 2.3.3 dates back to the second half of the nineteenth century and the studies of Heine and Borel on compact subsets of $\mathbb{R}$.

DEFINITION 2.3.1 Let $E$ be a subset of a metric space $(X, d)$. A collection $\left{O_{\alpha}\right}_{\alpha \in A}$ of open subsets of $X$ is an open cover of $E$ if
$$E \subset \bigcup_{\alpha \in A} O_{\alpha} .$$
An alternate definition is as follows: The collection $\left{O_{\alpha}\right}_{\alpha \in A}$ of open sets is an open cover of $E$ if for each $p \in E$, there exists an $\alpha \in A$ such that $p \in O_{\alpha}$.

EXAMPLES 2.3.2 (a) Let $E=(0,1)$ and $O_{n}=\left(0,1-\frac{1}{n}\right), n=2,3, \ldots$ Then $\left{O_{n}\right}_{n=2}^{\infty}$ is an open cover of $E$. To see this, suppose $x \in E$. Then since $x<1$, there exists an integer $n$ such that $x<1-1 / n$. Thus $x \in O_{n}$, and as a consequence
$$E \subset \bigcup_{n=2}^{\infty} O_{n},$$
which proves the assertion. In fact, since $O_{n} \subset E$ for each $n$, we have $E=$ $\bigcup_{n=2}^{\infty} O_{n}$
(b) Let $F=[0, \infty)$ and for each $n \in \mathbb{N}$ let $U_{n}=(-1, n)$. Then $\left{U_{n}\right}_{n \in \mathbb{N}}$ is an open cover of $F$.

## 数学代写|实分析作业代写Real analysis代考|Compact Subsets of R

We now turn to our goal of providing a characterization of the compact subsets of the real line $\mathbb{R}$. The first of the two main results is attributed to Eduard Heine (1821-1881) and Emile Borel (1871-1956), whereas the second is due to Bernhard Bolzano (1781-1848) and Karl Weierstrass (1815-1897). The two theorems rank very high among the many important advances in the foundations of analysis during the nineteenth century. The importance of these results will become evident in later chapters. As is to be expected, the least upper bound property of $\mathbb{R}$ will play a crucial role in the proofs of these theorems.

THEOREM 2.4.1 (Heine-Borel) Every closed and bounded interval $[a, b]$ is compact.
Proof. Let $\mathcal{U}=\left{U_{\alpha}\right}_{\alpha \in A}$ be an open cover of $[a, b]$ and let $E=\left{r \in[a, b]:[a, r]\right.$ is covered by a finite number of the sets $\left.U_{\alpha}\right}$.
The set $E$ is bounded above by $b$, and since $a \in U_{\alpha}$ for some $\alpha \in A, E$ is nonempty. Thus by the least upper bound property the supremum of $E$ exists in $\mathbb{R}$. Let $\alpha=\sup E$. Since $b$ is an upper bound of $E, \alpha \leq b$.

# 实分析代写

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

$$E \subset \bigcup_{\alpha \in A} O_{\alpha} .$$

$$E \subset \bigcup_{n=2}^{\infty} O_{n}$$

(b) 让 $F=[0, \infty)$ 并且对于每个 $n \in \mathbb{N}$ 让 $U_{n}=(-1, n)$. 然后 $\backslash$ left 的分隔符缺失或无法识别 是一个开盖 $F$.

## 有限元方法代写

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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。

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