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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 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|Sequences of Real Numbers

Now that we have covered the basic topological concepts required for the study of analysis, we begin with limits of sequences. This topic will be our first serious introduction to the limit process. The notion of convergence of a sequence dates back to the early nineteenth century and the work of Bolzano (1817) and Cauchy (1821). Some of the concepts and results included in this chapter have undoubtedly been encountered previously in the study of calculus. Our presentation however will be considerably more rigorous – emphasizing proofs rather than computations.

Although our primary emphasis will be on sequences of real numbers, these are not the only sequences which are typically encountered. It is not at all unusual to talk about sequences of functions, sequences of vectors, etc. For this reason we will begin our study of sequences in the general setting of metric spaces. Most of the examples however will come from the real numbers. A good understanding of sequences in $\mathbb{R}$ will prove helpful in providing insight into properties of sequences in more general settings.

We begin the chapter by introducing the notion of convergence of a sequence in a metric space, and then by proving the standard limit theorems for sequences of real numbers normally encountered in calculus. In Section $3.3$ we will use the least upper bound property of $\mathbb{R}$ to prove that every bounded monotone sequence of real numbers converges in $\mathbb{R}$. The study of subsequences and sub-sequential limits will be the topic of Section 3.4. In this section, we also prove the well known result of Bolzano and Weierstrass that every bounded sequence of real numbers has a convergent subsequence. This result will then be used to provide a short proof of the fact that every Cauchy sequence of real numbers converges. Although the study of series of real numbers is the main topic of Chapter 7 , some knowledge of series will be required in the construction of certain examples in Chapters 4 and 6 . For this reason we include a brief introduction to series as the last section of this chapter.

# 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 $\backslash 1$ eft 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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