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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|拓扑学代写Topology代考|CONVERGENCE, COMPLETENESS, AND BAIRE’S THEOREM

As we emphasized in the introduction to this chapter, one of our main aims in considering metric spaces is to study convergent sequences in a context more general than that of classical analysis. The fruits of this study are many, and among them is the added insight gained into ordinary convergence as it is used in analysis.
Let $X$ be a metric space with metric $d$, and let
$$\left{x_{n}\right}=\left{x_{1}, x_{2}, \ldots, x_{n}, \ldots\right}$$
be a sequence of points in $X$. We say that $\left{x_{n}\right}$ is convergent if there exists a point $x$ in $X$ such that either
(1) for each $\epsilon>0$, there exists a positive integer $n_{0}$ such that $n \geq n_{0} \Rightarrow d\left(x_{n}, x\right)<\epsilon$; or equivalently,
(2) for each open sphere $S_{\epsilon}(x)$ centered on $x$, there exists a positive integer $n_{0}$ such that $x_{n}$ is in $S_{c}(x)$ for all $n \geq n_{0}$.

The reader should observe that the first condition is a direct generalization of convergence for sequences of numbers as defined in the introduction, and that the second can be thought of as saying that each open sphere centered on $x$ contains all points of the sequence from some place on. If we rely on our knowledge of what is meant by a convergent sequence of real numbers, the statement that $\left{x_{n}\right}$ is convergent can equally well be defined as follows: there exists a point $x$ in $X$ such that $d\left(x_{n}, x\right) \rightarrow 0$. We usually symbolize this by writing
$$x_{n} \rightarrow x,$$
and we express it verbally by saying that $x_{n}$ approaches $x$, or that $x_{n}$ converges to $x$. It is easily seen from condition (2) and Problem 10-1 that the point $x$ in this discussion is unique, that is, that $x_{n} \rightarrow y$ with $y \neq x$ is impossible. The point $x$ is called the limit of the sequence $\left{x_{n}\right}$, and we sometimes write $x_{n} \rightarrow x$ in the form
$$\lim x_{n}=x \text {. }$$

## 数学代写|拓扑学代写Topology代考|CONTINUOUS MAPPINGS

In the previous section we extended the idea of convergence to the context of a general metric space. We now do the same for continuity.
Let $X$ and $Y$ be metric spaces with metrics $d_{1}$ and $d_{2}$, and let $f$ be a mapping of $X$ into $Y$. $f$ is said to be continuous at a point $x_{0}$ in $X$ if either
‘There is some rather undescriptive terminology which is of en used in connection with Baire’s theorem. We shall not make use of it ourselves, but the reader ought to be acquainted with it. A subset of a metric space is called a set of the first category if it can be represented as the union of a sequence of nowhere dense sets, and a set of the second calegory if it is not a set of the first category. Baire’s theorem-sometimes called the Baire category theorem-can now be expressed as follows: any complete metric space (considered as a subset of itself) is a set of the second category.

of the following equivalent conditions is satisfied:
(1) for each $\epsilon>0$ there exists $\delta>0$ such that $d_{1}\left(x, x_{0}\right)<\delta \Rightarrow$ $d_{2}\left(f(x), f\left(x_{0}\right)\right)<\epsilon$
(2) for each open sphere $S_{\epsilon}\left(f\left(x_{0}\right)\right.$ ) centered on $f\left(x_{0}\right)$ there exists an open sphere $S_{\delta}\left(x_{0}\right)$ centered on $x_{0}$ such that $f\left(S_{\delta}\left(x_{0}\right)\right) \subseteq S_{\epsilon}\left(f\left(x_{0}\right)\right)$.
The reader will notice that the first condition generalizes the elementary definition given in the introduction to this chapter, and that the second translates the first into the language of open spheres.

Our first theorem expresses continuity at a point in terms of sequences which converge to the point.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考| CONVERGENCE, COMPLETENESS, AND BAIRE’S THEOREM

1) 对于每个 $\epsilon>0$ ，则存在一个正整数 $n_{0}$ 使得 $n \geq n_{0} \Rightarrow d\left(x_{n}, x\right)<\epsilon$; 或等价地，
(2) 对于每个开放球体 $S_{\epsilon}(x)$ 以 $x$ ，则存在一个正整数 $n_{0}$ 使得 $x_{n}$ 位于 $S_{c}(x)$ 面向所有人 $n \geq n_{0}$.

$$x_{n} \rightarrow x$$

$$\lim x_{n}=x .$$

## 数学代写|拓扑学代写Topology代考| CONTINUOUS MAPPINGS

(1) 每个 $\epsilon>0$ 存在 $\delta>0$ 使得 $d_{1}\left(x, x_{0}\right)<\delta \Rightarrow d_{2}\left(f(x), f\left(x_{0}\right)\right)<\epsilon$
(2) 对于每个开放球体 $S_{\epsilon}\left(f\left(x_{0}\right)\right)$ 的中心位于 $f\left(x_{0}\right)$ 存在一个开放的球体 $S_{\delta}\left(x_{0}\right)$ 以 $x_{0}$ 使得 $f\left(S_{\delta}\left(x_{0}\right)\right) \subseteq S_{\epsilon}\left(f\left(x_{0}\right)\right)$.

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师