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## 数学代写|实分析代写Real Analysis代考|Completion of a Measure Space

If $(X, \mathcal{A}, \mu)$ is a measure space, we define the completion of this space to be the measure space $(X, \overline{\mathcal{A}}, \bar{\mu})$ defined by
\begin{aligned} \overline{\mathcal{A}} & =\left{\begin{array}{l|l} E \Delta Z & \begin{array}{l} E \text { is in } \mathcal{A} \text { and } Z \subseteq Z^{\prime} \text { for } \ \text { some } Z^{\prime} \in \mathcal{A} \text { with } \mu\left(Z^{\prime}\right)=0 \end{array} \end{array},\right. \ \bar{\mu}(E \Delta Z) & =\mu(E) . \end{aligned}
It is necessary to verify that the result is in fact a measure space, and we shall carry out this step in the proposition below. In the case of Lebesgue measure $m$ on the line, when initially defined on the $\sigma$-algebra $\mathcal{A}$ of Borel sets, the sets in $\sigma$-algebra $\overline{\mathcal{A}}$ are said to be Lebesgue measurable.

Proposition 5.38. If $(X, \mathcal{A}, \mu)$ is a measure space, then the completion $(X, \overline{\mathcal{A}}, \bar{\mu})$ is a measure space. Specifically
(a) $\overline{\mathcal{A}}$ is a $\sigma$-algebra containing $\mathcal{A}$,
(b) the set function $\bar{\mu}$ is unambiguously defined on $\overline{\mathcal{A}}$, i.e., if $E_1 \Delta Z_1=$ $E_2 \Delta Z_2$ as above, then $\mu\left(E_1\right)=\mu\left(E_2\right)$,
(c) $\bar{\mu}$ is a measure on $\overline{\mathcal{A}}$, and $\bar{\mu}(E)=\mu(E)$ for all sets $E$ in $\mathcal{A}$.

(d) if $\tilde{\mu}$ is any measure on $\overline{\mathcal{A}}$ such that $\tilde{\mu}(E)=\mu(E)$ for all $E$ in $\mathcal{A}$, then $\tilde{\mu}=\bar{\mu}$ on $\overline{\mathcal{A}}$
(e) if $\mu(X)<+\infty$ and if for $E \subseteq X, \mu_(E)$ and $\mu^(E)$ are defined by $$\mu_(E)=\sup {A \subseteq E, A \in \mathcal{A}} \mu(A) \quad \text { and } \quad \mu^(E)=\inf {A \supseteq E, A \in \mathcal{A}} \mu(A) \text {, }$$
then $E$ is in $\overline{\mathcal{A}}$ if and only if $\mu_(E)=\mu^(E)$.
Proof. For (a), certainly $\mathcal{A} \subseteq \overline{\mathcal{A}}$ because we can use $Z=Z^{\prime}=\varnothing$ in the definition of $\overline{\mathcal{A}}$. Since $(E \Delta Z)^c=(E \Delta Z) \Delta X=(E \Delta X) \Delta Z=E^c \Delta Z, \overline{\mathcal{A}}$ is closed under complements.
To prove closure under countable unions, let us first prove that
$$\overline{\mathcal{A}}=\left{\begin{array}{l|l} E \cup Z & \begin{array}{l} E \text { is in } \mathcal{A} \text { and } Z \subseteq Z^{\prime} \text { for } \ \text { some } Z^{\prime} \in \mathcal{A} \text { with } \mu\left(Z^{\prime}\right)=0 \end{array} \end{array}\right} .$$

## 数学代写|实分析代写Real Analysis代考|Fubini’s Theorem for the Lebesgue Integral

Fubini’s Theorem for the Lebesgue integral concerns the interchange of order of integration of functions of two variables, just as with the Riemann integral in Section III.9. In the case of Euclidean space $\mathbb{R}^n$, we could have constructed Lebesgue measure in each dimension by a procedure similar to the one we used for $\mathbb{R}^1$. Then Fubini’s Theorem relates integration of a function of $m+n$ variables over a set by either integrating in all variables at once or integrating in the first $m$ variables first or integrating in the last $n$ variables first. In the context of more general measure spaces, we need to develop the notion of the product of two measure spaces. This corresponds to knowing $\mathbb{R}^m$ and $\mathbb{R}^n$ with their Lebesgue measures and to constructing $\mathbb{R}^{m+n}$ with its Lebesgue measure.

In the theorem as we shall state it, we are given two measures spaces $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, v)$, and we assume that both $\mu$ and $v$ are $\sigma$-finite. We shall construct a product measure space $(X \times Y, \mathcal{A} \times \mathcal{B}, \mu \times v)$, and the formula in question will be
\begin{aligned} \int_{X \times Y} f d(\mu \times v) & \stackrel{?}{=} \int_X\left[\int_Y f(x, y) d v(y)\right] d \mu(x) \ & \stackrel{?}{=} \int_Y\left[\int_X f(x, y) d \mu(x)\right] d v(y) . \end{aligned}
This formula will be valid for $f \geq 0$ measurable with respect to $\mathcal{A} \times \mathcal{B}$.
The technique of proof will be the standard one indicated in connection with proving Corollary 5.28. We start with indicator functions, extend the result to simple functions by linearity, and pass to the limit by the Monotone Convergence Theorem (Theorem 5.25). It is then apparent that the difficult step is the case that $f$ is an indicator function. In fact, it is not even clear in this special case that the inside integral $\int_Y I_E(x, y) d v(y)$ is a measurable function of $X$, and this is the step that requires some work.

## 数学代写|实分析代写Real Analysis代考|Completion of a Measure Space

\begin{aligned} \overline{\mathcal{A}} & =\left{\begin{array}{l|l} E \Delta Z & \begin{array}{l} E \text { is in } \mathcal{A} \text { and } Z \subseteq Z^{\prime} \text { for } \ \text { some } Z^{\prime} \in \mathcal{A} \text { with } \mu\left(Z^{\prime}\right)=0 \end{array} \end{array},\right. \ \bar{\mu}(E \Delta Z) & =\mu(E) . \end{aligned}

(a) $\overline{\mathcal{A}}$是包含$\mathcal{A}$的$\sigma$ -代数，

## 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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## 数学代写|实分析代写Real Analysis代考|Definition and Properties of Riemann Integral

The present section extends that development to several variables. A certain amount of the theory parallels what happened in one variable, and proofs for that part of the theory can be obtained by adjusting the notation and words of Section I.4 in simple ways. Results of that kind are much of the subject matter of this section.

In later sections we shall take up results having no close analog in Section I.4. The main results of this kind are
(i) a necessary and sufficient condition for a function to be Riemann integrable,
(ii) Fubini’s Theorem, concerning the relationship between multiple integrals and iterated integrals in the various possible orders,
(iii) a change-of-variables formula for multiple integrals.
We begin a discussion of these in the next section.
The one-variable theory worked with a bounded function $f:[a, b] \rightarrow \mathbb{R}$, with domain a closed bounded interval, and we now work with a bounded function $f: A \rightarrow \mathbb{R}$ with domain $A$ a “closed rectangle” in $\mathbb{R}^n$. For this purpose a closed rectangle (or “closed geometric rectangle”) in $\mathbb{R}^n$ is a bounded set of the form
$$A=\left[a_1, b_1\right] \times \cdots \times\left[a_n, b_n\right]$$
with $a_j \leq b_j$ for all $j$. Let us abbreviate $\left[a_j, b_j\right]$ as $A_j$. In geometric terms the sides or faces are assumed parallel to the axes or coordinate hyperplanes. We shall use the notion of open rectangle in later sections and chapters, an open rectangle being a similar product of bounded open intervals $\left(a_j, b_j\right)$ for $1 \leq j \leq n$. However, in this section the term “rectangle” will always mean closed rectangle.

## 数学代写|实分析代写Real Analysis代考|Riemann Integrable Functions

Let $E$ be a subset of $\mathbb{R}^n$. We say that $E$ is of measure 0 if for any $\epsilon>0, E$ can be covered by a finite or countably infinite set of closed rectangles in the sense of Section 7 of total volume less than $\epsilon$. It is equivalent to require that $E$ can be covered by a finite or countably infinite set of open rectangles of total volume less than $\epsilon$. In fact, if a system of open rectangles covers $E$, then the system of closures covers $E$ and has the same total volume; conversely if a system of closed rectangles covers $E$, then the system of open rectangles with the same centers and with sides expanded by a factor $1+\delta$ covers $E$ as long as $\delta>0$.

Several properties of sets of measure 0 are evident: a set consisting of one point is of measure 0 , a face of a closed rectangle is a set of measure 0 , and any subset of a set of measure 0 is of measure 0 . Less evident is the fact that the countable union of sets of measure 0 is of measure 0 . In fact, if $\epsilon>0$ is given and if $E_1, E_2, \ldots$ are sets of measure 0 , find finite or countably infinite systems $\mathcal{R}_j$ of closed rectangles for $j \geq 1$ such that the total volume of the members of $\mathcal{R}_j$ is $<\epsilon / 2^n$. Then $\mathcal{R}=\bigcup_j \mathcal{R}_j$ is a system of closed rectangles covering $\bigcup_j E_j$ and having total volume $<\epsilon$.

The goal of this section is to prove the following theorem, which gives a useful necessary and sufficient condition for a function of several variables to be Riemann integrable. The theorem immediately extends from the scalar-valued case as stated to the case that $f$ has values in $\mathbb{R}^m$ or $\mathbb{C}^m$.

## 数学代写|实分析代写Real Analysis代考|Definition and Properties of Riemann Integral

(1)函数为Riemann可积的充分必要条件;
(ii)关于多重积分和不同阶次的迭代积分之间关系的富比尼定理;
(iii)多重积分的变量变换公式。

$$A=\left[a_1, b_1\right] \times \cdots \times\left[a_n, b_n\right]$$

## 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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## 数学代写|实分析代写Real Analysis代考|Baire Category Theorem

A number of deep results in analysis depend critically on the fact that some metric space is complete. Already we have seen that the metric space $C(S)$ of bounded continuous scalar-valued functions on a metric space is complete, and we shall see as not too hard a consequence in Chapter XII that there exists a continuous periodic function whose Fourier series diverges at a point. One of the features of the Lebesgue integral in Chapter $\mathrm{V}$ will be that the metric spaces of integrable functions and of square-integrable functions, with their natural metrics, are further examples of complete metric spaces. Thus these spaces too are available for applications that make use of completeness.

The main device through which completeness is transformed into a powerful hypothesis is the Baire Category Theorem below. A closed set in a metric space is nowhere dense if its interior is empty. Its complement is an open dense set, and conversely the complement of any open dense set is closed nowhere dense.
EXAMPLE. A nontrivial example of a closed nowhere dense set is a Cantor set ${ }^4$ in $\mathbb{R}$. This is a set constructed from a closed bounded interval of $\mathbb{R}$ by removing an open interval in the middle of length a fraction $r_1$ of the total length with $0<r_1<1$, removing from each of the 2 remaining closed subintervals an open interval in the middle of length a fraction $r_2$ of the total length of the subinterval, removing from each of the 4 remaining closed subintervals an open interval in the middle of length a fraction $r_3$ of the total length of the interval, and so on indefinitely. The Cantor set is obtained as the intersection of the approximating sets. It is closed, being the intersection of closed sets, and it is nowhere dense because it contains no interval of more than one point. For the standard Cantor set, the starting interval is $[0,1]$, and the fractions are given by $r_1=r_2=\cdots=\frac{1}{3}$ at every stage. In general, the “length” of the resulting $\operatorname{set}^5$ is the product of the length of the starting interval and $\prod_{n=1}^{\infty}\left(1-r_n\right)$.

## 数学代写|实分析代写Real Analysis代考|Properties of C(S) for Compact Metric S

If $(S, d)$ is a metric space, then we saw in Proposition 2.44 that the vector space $B(S)$ of bounded scalar-valued functions on $S$, in the uniform metric, is a complete metric space. We saw also in Corollary 2.45 that the vector subspace $C(S)$ of bounded continuous functions is a complete subspace. In this section we shall study the space $C(S)$ further under the assumption that $S$ is compact. In this case Propositions 2.38 and 2.34 tell us that every continuous scalar-valued function on $S$ is automatically bounded and hence is in $C(S)$.

The first result about $C(S)$ for $S$ compact is a generalization of Ascoli’s Theorem from its setting in Theorem 1.22 for real-valued functions on a bounded interval $[a, b]$. The generalized theorem provides an insight that is not so obvious from the special case that $S$ is a closed bounded interval of $\mathbb{R}$. The insight is a characterization of the compact subsets of $C(S)$ when $S$ is compact, and it is stated precisely in Corollary 2.57 below. The relevant definitions for Ascoli’s Theorem are generalized in the expected way. Let $\mathcal{F}=\left{f_\alpha \mid \alpha \in A\right}$ be a set of scalar-valued functions on the compact metric space $S$. We say that $\mathcal{F}$ is equicontinuous at $x \in S$ if for each $\epsilon>0$, there is some $\delta>0$ such that $d(t, x)<\delta$ implies $|f(t)-f(x)|<\epsilon$ for all $f \in \mathcal{F}$. The set $\mathcal{F}$ of functions is pointwise bounded if for each $t \in[a, b]$, there exists a number $M_t$ such that $|f(t)| \leq M_t$ for all $f \in \mathcal{F}$. The set is uniformly equicontinuous on $S$ if it is equicontinuous at each point $x \in S$ and if the $\delta$ can be taken independent of $x$. The set is uniformly bounded on $S$ if it is pointwise bounded at each $t \in S$ and the bound $M_t$ can be taken independent of $t$; this last definition is consistent with the definition of a uniformly bounded sequence of functions given in Section 4.
Theorem 2.56 (Ascoli’s Theorem). Let $(S, d)$ be a compact metric space. If $\left{f_n\right}$ is a sequence of scalar-valued functions on $S$ that is equicontinuous at each point of $S$ and pointwise bounded on $S$, then
(a) $\left{f_n\right}$ is uniformly equicontinuous and uniformly bounded on $S$,
(b) $\left{f_n\right}$ has a uniformly convergent subsequence.

## 数学代写|实分析代写Real Analysis代考|Properties of C(S) for Compact Metric S

(a) $\left{f_n\right}$在$S$上均匀等连续，均匀有界;
(b) $\left{f_n\right}$具有一致收敛的子序列。

## 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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## 数学代写|实分析代写Real Analysis代考|Definition and Examples

Let $X$ be a nonempty set. A function $d$ from $X \times X$, the set of ordered pairs of members of $X$, to the real numbers is a metric, or distance function, if
(i) $d(x, y) \geq 0$ always, with equality if and only if $x=y$,
(ii) $d(x, y)=d(y, x)$ for all $x$ and $y$ in $X$,
(iii) $d(x, y) \leq d(x, z)+d(z, y)$ for all $x, y$, and $z$, the triangle inequality. In this case the pair $(X, d)$ is called a metric space.

The real line $\mathbb{R}^1$ with metric $d(x, y)=|x-y|$ is the motivating example. Properties (i) and (ii) are apparent, and property (iii) is readily verified one case at a time according as $z$ is less than both $x$ and $y, z$ is between $x$ and $y$, or $z$ is greater than both $x$ and $y$.

We come to further examples in a moment. Particularly in the case that $X$ is a space of functions, a space may turn out to be almost a metric space but not to satisfy the condition that $d(x, y)=0$ implies $x=y$. Accordingly we introduce a weakened version of (i) as
(i’) $d(x, y) \geq 0$ and $d(x, x)=0$ always,
and we say that a function $d$ from $X \times X$ to the real numbers is a pseudometric if (i’), (ii), and (iii) hold. In this case, $(X, d)$ is called a pseudometric space.
Let $(X, d)$ be a pseudometric space. If $r>0$, the open ball of radius $r$ and center $x$, denoted by $B(r ; x)$, is the set of points at distance less than $r$ from $x$, namely
$$B(r ; x)={y \in X \mid d(x, y)<r} .$$

## 数学代写|实分析代写Real Analysis代考|Open Sets and Closed Sets

In this section we generalize the Euclidean notions of open set, closed set, neighborhood, interior, limit point, and closure so that they make sense for all pseudometric spaces, and we prove elementary properties relating these metricspace notions. In working with metric spaces and pseudometric spaces, it is often helpful to draw pictures as if the space in question were $\mathbb{R}^2$, even computing distances that are right for $\mathbb{R}^2$. We shall do that in the case of the first lemma but not afterward in this section. Let $(X, d)$ be a pseudometric space.

Lemma 2.4. If $z$ is in the intersection of open balls $B(r ; x)$ and $B(s ; y)$, then there exists some $t>0$ such that the open ball $B(t ; z)$ is contained in that intersection. Consequently the intersection of two open balls is open.

REMARK. Figure 2.2 shows what $B(t ; z)$ looks like in the metric space $\mathbb{R}^2$.
PROOF. Take $t=\min {r-d(x, z), s-d(y, z)}$. If $w$ is in $B(t ; z)$, then the triangle inequality gives
$$d(x, w) \leq d(x, z)+d(z, w)<d(x, z)+t \leq d(x, z)+(r-d(x, z))=r,$$
and hence $w$ is in $B(r ; x)$. Similarly $w$ is in $B(s ; y)$.

Proposition 2.5. The open sets of $X$ have the properties that
(a) $X$ and the empty set $\varnothing$ are open,
(b) an arbitrary union of open sets is open,
(c) any finite intersection of open sets is open.
PROOF. We know from Lemma 2.1 that a set is open if and only if it is the union of open balls. Then (b) is immediate, and (a) follows, since $X$ is the union of all open balls and $\varnothing$ is an empty union. For (c), it is enough to prove that $U \cap V$ is open if $U$ and $V$ are open. Write $U=\bigcup_\alpha B_\alpha$ and $V=\bigcup_\beta B_\beta$ as unions of open balls. Then $U \cap V=\bigcup_{\alpha, \beta}\left(B_\alpha \cap B_\beta\right)$, and Lemma 2.4 shows that $U \cap V$ is exhibited as the union of open balls. Thus $U \cap V$ is open.

## 数学代写|实分析代写Real Analysis代考|Definition and Examples

(i) $d(x, y) \geq 0$总是，当且仅当$x=y$，
(ii) $X$中所有的$x$和$y$为$d(x, y)=d(y, x)$;
(iii) $d(x, y) \leq d(x, z)+d(z, y)$对于所有的$x, y$，和$z$，三角不等式。在这种情况下$(X, d)$对被称为度量空间。

(i’)$d(x, y) \geq 0$和$d(x, x)=0$总是，

$$B(r ; x)={y \in X \mid d(x, y)<r} .$$

## 数学代写|实分析代写Real Analysis代考|Open Sets and Closed Sets

$$d(x, w) \leq d(x, z)+d(z, w)<d(x, z)+t \leq d(x, z)+(r-d(x, z))=r,$$

(a) $X$和空集$\varnothing$打开;
(b)开集的任意并是开的;
(c)任意开集的有限交是开的。

## 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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## 数学代写|实分析代写Real Analysis代考|Interchange of Limits

Let $\left{b_{i j}\right}$ be a doubly indexed sequence of real numbers. It is natural to ask for the extent to which
$$\lim i \lim _j b{i j}=\lim j \lim _i b{i j}$$
more specifically to ask how to tell, in an expression involving iterated limits, whether we can interchange the order of the two limit operations. We can view matters conveniently in terms of an infinite matrix
$$\left(\begin{array}{ccc} b_{11} & b_{12} & \cdots \ b_{21} & b_{22} & \ \vdots & & \ddots \end{array}\right)$$
The left-hand iterated limit, namely $\lim i \lim _j b{i j}$, is obtained by forming the limit of each row, assembling the results, and then taking the limit of the row limits down through the rows. The right-hand iterated limit, namely $\lim j \lim _i b{i j}$, is obtained by forming the limit of each column, assembling the results, and then taking the limit of the column limits through the columns. If we use the particular infinite matrix
$$\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & \cdots \ 0 & 1 & 1 & 1 & \cdots \ 0 & 0 & 1 & 1 & \cdots \ 0 & 0 & 0 & 1 & \cdots \ \vdots & & & & \ddots \end{array}\right)$$
then we see that the first iterated limit depends only on the part of the matrix above the main diagonal, while the second iterated limit depends only on the part of the matrix below the main diagonal. Thus the two iterated limits in general have no reason at all to be related. In the specific matrix that we have just considered, they are 1 and 0 , respectively. Let us consider some examples along the same lines but with an analytic flavor.

## 数学代写|实分析代写Real Analysis代考|Uniform Convergence

Let us examine more closely what is happening in the proof of Theorem 1.13 , in which it is proved that iterated limits can be interchanged under certain hypotheses of monotonicity. One of the iterated limits is $L=\lim i \lim _j b{i j}$, and the claim is that $L$ is approached as $i$ and $j$ tend to infinity jointly. In terms of a matrix whose entries are the various $b_{i j}$ ‘s, the pictorial assertion is that all the terms far down and to the right are close to $L$ :
$$\left(\begin{array}{cc} \cdots & \cdots \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L \end{array} \end{array}\right) \text {. }$$
To see this claim, let us choose a row limit $L_{i_0}$ that is close to $L$ and then take an entry $b_{i_0 j_0}$ that is close to $L_{i_0}$. Then $b_{i_0 j_0}$ is close to $L$, and all terms down and to the right from there are even closer because of the hypothesis of monotonicity.
To relate this behavior to something uniform, suppose that $L<+\infty$, and let some $\epsilon>0$ be given. We have just seen that we can arrange to have $\left|L-b_{i j}\right|<\epsilon$ whenever $i \geq i_0$ and $j \geq j_0$. Then $\left|L_i-b_{i j}\right|<\epsilon$ whenever $i \geq i_0$, provided $j \geq j_0$. Also, we have $\lim j b{i j}=L_i$ for $i=1,2, \ldots, i_0-1$. Thus $\left|L_i-b_{i j}\right|<\epsilon$ for all $i$, provided $j \geq j_0^{\prime}$, where $j_0^{\prime}$ is some larger index than $j_0$. This is the notion of uniform convergence that we shall define precisely in a moment: an expression with a parameter ( $j$ in our case) has a limit (on the variable $i$ in our case) with an estimate independent of the parameter. We can visualize matters as in the following matrix:
$$i\left(\begin{array}{l|l} j & j_0^{\prime} \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L_i \ \text { on all rows. } \end{array} \end{array}\right) .$$
The vertical dividing line occurs when the column index $j$ is equal to $j_0^{\prime}$, and all terms to the right of this line are close to their respective row limits $L_i$.

## 数学代写|实分析代写Real Analysis代考|Interchange of Limits

$$\lim i \lim j b{i j}=\lim j \lim _i b{i j}$$ 更具体地说，是问如何判断，在一个包含迭代极限的表达式中，我们是否可以交换两个极限运算的顺序。我们可以方便地用无穷矩阵的形式来考虑问题 $$\left(\begin{array}{ccc} b{11} & b_{12} & \cdots \ b_{21} & b_{22} & \ \vdots & & \ddots \end{array}\right)$$

$$\left(\begin{array}{ccccc} 1 & 1 & 1 & 1 & \cdots \ 0 & 1 & 1 & 1 & \cdots \ 0 & 0 & 1 & 1 & \cdots \ 0 & 0 & 0 & 1 & \cdots \ \vdots & & & & \ddots \end{array}\right)$$

## 数学代写|实分析代写Real Analysis代考|Uniform Convergence

$$\left(\begin{array}{cc} \cdots & \cdots \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L \end{array} \end{array}\right) \text {. }$$

$$i\left(\begin{array}{l|l} j & j_0^{\prime} \ \cdots & \begin{array}{l} \text { All terms here } \ \text { are close to } L_i \ \text { on all rows. } \end{array} \end{array}\right) .$$

## 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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## 数学代写|实分析代写Real Analysis代考|Integration by substitution

Let (i) $f(a, b]-\mathbb{R}$ be integrable on $[a, b]$, $a \phi(\beta)=b, a n d$
(iii) $f \circ \phi$ and $\phi^{\prime}$ are integrable on $[\alpha, \beta]$ and $\phi^{\prime}(t) \neq 0$ for all $t, \theta$ $[\alpha, \beta]$
Then $\int_a^b f(x) d x=\int_\alpha^\beta f(\phi(t)) \phi^{\prime}(t) d t$
Since $\phi^{\prime}(t) \neq 0$ on $[\alpha, \beta]$, it follows from Darboux’s theorem that either $\phi^{\prime}(t)>0$ for all $t \in[\alpha, \beta]$ or $\phi^{\prime}(t)<0$ for all $t \in[\alpha, \beta]$, i.e., either $\phi$ is strictly increasing on $[\alpha, \beta]$ or $\phi$ is strictly decreasing on $[\alpha, \beta]$.
Accordingly, the theorem can be stated in two parts.
First part.
Let (i) $f:[a, b] \rightarrow \mathbb{R}$ be integrable on $[a, b]$,
(ii) $\phi:[\alpha, \beta] \rightarrow \mathbb{R}$ be differentiable and strictly increasing on $[\alpha, \beta]$ such that $\phi(\alpha)=a, \phi(\beta)=b$, and
(iii) $f \circ \phi$ and $\phi^{\prime}$ are integrable on $[\alpha, \beta]$.
Then $\int_a^b f(x) d x=\int_\alpha^\beta f(\phi(t)) \phi^{\prime}(t) d t$.
Proof. Since $\phi$ is differentiable on $[\alpha, \beta], \phi$ is continuous on $[\alpha, \beta]$.
Since $\phi$ is continuous and strictly increasing on $[\alpha, \beta]$ and $\phi(\alpha)=$ $a, \phi(\beta)=b, \phi^{-1}$ is continuous and strictly increasing on $[a, b]$.

Let $P=\left(x_0, x_1, \ldots, x_n\right)$ be any partition of $[a, b]$ and $Q=$ $\left{y_0, y_1, \ldots, y_n\right}$ where $y_i=\phi^{-1}\left(x_i\right)$ be the corresponding partition of $[\alpha, \beta]$.

By Lagrange’s Mean value theorem for the function $\phi$ on $\left[y_{r-1}, y_r\right]$, $\phi\left(y_r\right)-\phi\left(y_{r-1}\right)=\left(y_r-y_{r-1}\right) \phi^{\prime}\left(r_r\right)$ for some $r_r \in\left(y_{r-1}, y_r\right)$.
That is, $x_r-x_{r-1}=\left(y_r-y_{r-1}\right) \phi^{\prime}\left(r_r\right), r=1,2, \ldots, n \ldots$
Let $\phi\left(\eta_r\right)=\xi_r, r=1,2, \ldots, n$.
Now $S(P, f, \xi)=f\left(\xi_1\right)\left(x_1-x_0\right)+f\left(\xi_2\right)\left(x_2-x_1\right)+\cdots+f\left(\xi_n\right)\left(x_n-x_{n-1}\right)$ $=f\left(\phi\left(\eta_1\right)\right) \phi^{\prime}\left(\eta_1\right)\left(y_1-y_0\right)+\cdots+f\left(\phi\left(\eta_n\right)\right) \phi^{\prime}\left(\eta_n\right)\left(y_n-y_{n-1}\right)$ $=S\left(Q,(f \circ \phi) \cdot \phi^{\prime}, \eta\right)$.
Since $f$ is integrable on $[a, b], \lim _{|P| \rightarrow 0} S(P, f, \xi)=\int_a^b f$.

## 数学代写|实分析代写Real Analysis代考|Integration by parts

Theorem 11.11.1. Let $f:[a, b] \rightarrow \mathbb{R}, g:[a, b] \rightarrow \mathbb{R}$ be both differentiable on $[a, b]$ and $f^{\prime}, g^{\prime}$ are both integrable on $[a, b]$. Then
$$\int_a^b f(x) g^{\prime}(x) d x=f(b) g(b)-f(a) g(a)-\int_a^b f^{\prime}(x) g(x) d x .$$
Proof. Since $f$ and $g$ are both differentiable on $[a, b], f g$ is differentiable on $[a, b]$.

Since $f$ and $g$ are differentiable on $[a, b], f$ and $g$ are continuous on $[a, b]$ and therefore they are both integrable on $[a, b]$.

Therefore $f g^{\prime}+f^{\prime} g$ is integrable on $[a, b]$, i.e., $(f g)^{\prime}$ is integrable on ${a, b]$.

So by the fundamental theorem, $\int_a^b\left(f^{\prime} g\right)^{\prime}=[f g]_a^b=f(b) g(b)-$ $f(a) g(a)$. Also $\int_a^b(f g)^{\prime}=\int_a^b\left(f g^{\prime}+f^{\prime} g\right)=\int_a^b f g^{\prime}+\int_a^b f^{\prime} g$.
Therefore $\int_a^b f(x) g^{\prime}(x) d x+\int_a^b f^{\prime}(x) g(x) d x=f(b) g(b)-f(a) g(a)$ or, $\int_a^b f(x) g^{\prime}(x) d x=f(b) g(b)-f(a) g(a)-\int_a^b f^{\prime}(x) g(x) d x$.

Theorem 11.12.1. (First Mean value theorem)
If (i) $f:[a, b] \rightarrow \mathbb{R}$ and $g:[a, b] \rightarrow \mathbb{R}$ be both integrable on $[a, b]$, and
(ii) $g(x)$ has the same sign for all $x \in[a, b]$,
then there is a number $\mu$ such that
$\int_a^b f(x) g(x) d x=\mu \int_a^b g(x) d x$ where $m \leq \mu \leq M$ and
$$m=\inf {x \in[a, b]} f(x), M=\sup {x \in[a, b]} g(x) \text {. }$$
If further, $f$ is continuous on $[a, b]$ then there exists a point $\xi$ in $[a, b]$ such that $\int_a^b f(x) g(x) d x=f(\xi) \int_a^b g(x) d x$.
Proof. Case 1. Let $g(x)>0, x \in[a, b]$.
Since $m=\inf {x \in[a, b]} f(x)$ and $M=\sup {x \in[a, b]} f(x), m \leq f(x) \leq M$ for all $x \in[a, b]$. Therefore $m g(x) \leq f(x) g(x) \leq M g(x)$ for all $x \in[a, b]$.

Since $f$ and $g$ are both integrable on $[a, b], m g, f g$ and $M g$ are integrable on $[a, b]$, and
\begin{aligned} & \int_n^b m g(x) d x \leq \int_a^b f(x) g(x) d x \leq \int_a^b M g(x) d x \ & \text { or, } m \int_a^b g(x) d x \leq \int_a^b f(x) g(x) d x \leq M \int_a^b g(x) d x . \end{aligned}
Therefore $\int_a^b f(x) g(x) d x=\mu \int_a^b g(x) d x$, where $m \leq \mu \leq M$.

## 数学代写|实分析代写Real Analysis代考|Integration by substitution

(iii) $f \circ \phi$ 和 $\phi^{\prime}$ 是可积的 $[\alpha, \beta]$ 和 $\phi^{\prime}(t) \neq 0$ 对所有人 $t, \theta$ $[\alpha, \beta]$

(ii) $\phi:[\alpha, \beta] \rightarrow \mathbb{R}$ 是可微且严格递增的 $[\alpha, \beta]$ 这样 $\phi(\alpha)=a, \phi(\beta)=b$，和
(iii) $f \circ \phi$ 和 $\phi^{\prime}$ 是可积的 $[\alpha, \beta]$．

(ii) $g(x)$对所有$x \in[a, b]$具有相同的符号，

$\int_a^b f(x) g(x) d x=\mu \int_a^b g(x) d x$其中$m \leq \mu \leq M$和
$$m=\inf {x \in[a, b]} f(x), M=\sup {x \in[a, b]} g(x) \text {. }$$

\begin{aligned} & \int_n^b m g(x) d x \leq \int_a^b f(x) g(x) d x \leq \int_a^b M g(x) d x \ & \text { or, } m \int_a^b g(x) d x \leq \int_a^b f(x) g(x) d x \leq M \int_a^b g(x) d x . \end{aligned}

## 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。

Posted on Categories:Real analysis, 实分析, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|实分析代写Real Analysis代考|Some Riemann integrable functions

Theorem 11.5,1. Let a functionfs, $[a, b] \rightarrow \mathbb{R}$ be nonatotut $[a, b]$. Then $f$ is intagrable on $[a, b]$.
Proof. Let $f$ be monotone increasing on $[a, b]$. Clearly, $f$ is bounded on $[a, b], f(a)$ being a lower bound and $f(b)$ being an upper bound of $f$ on $[a, b] . f(b)-f(a) \geq 0$
Let us choose $\epsilon>0$.
Let $P$ be a partition of $[a, b]$ with $|P|<\epsilon /{f(b)-f(a)+1}$. Let $P=\left(x_0, x_1, x_2, \ldots, x_n\right)$, where $a=x_0<x_1<x_2<\cdots<x_n=b$.

Let $M_r=\sup {x \in\left[x{r-1}, x_r\right]} f(x), \pi m_r=\inf {x \in\left[x{r-1}, x_r\right]} f(x)$, for $r=1,2, \ldots, n$.
Then $M_r=f\left(x_r\right)$ and $m_r=f\left(x_{r-1}\right)$, for $r=1,2, \ldots, n$.

\text { We have } \begin{aligned} U(P, f)-L(P, f) & =\sum_{r=1}^n\left(M_r-m_r\right)\left(x_r-x_{r-1}\right) \ & =\sum_{r=1}^n\left{f\left(x_r\right)-f\left(x_{r-1}\right)\right}\left(x_r-x_{r-1}\right) \ & \leq|P| \sum_{r=1}^n\left{f\left(x_r\right)-f\left(x_{r-1}\right)\right} \ & =|P|{f(b)-f(a)}<\epsilon . \end{aligned}
Therefore for a chosen positive $\epsilon$, there exists a partition $P$ of $[a, b]$ such that $U(P, f)-L(P, f)<\epsilon$.

This being a suffientandion for integrability, $f$ – is integrable on $[a, b]$.

Proceeding in a similar manner it can be proved that if $f$ be monotone decreasing on $[a, b]$, then $f$ is integrable on $[a, b]$.

## 数学代写|实分析代写Real Analysis代考|Properties of Riemann integrable functions.

Theorem 11 6.1, Lét $f:[a, b] \rightarrow \mathbb{R}, g:[a, b] \rightarrow \mathbb{R}$ be both integrable on $[a, b]$. Then $f+g$ ts integrable on ${a, b} a$ and $f_a^b(f+g)=\int_a^b f_f f_a^b g$.
Proof Since $f \in \mathcal{R}[a, b]$ and $g \in \mathcal{R}[a, b], f$ and $g$ are both bounded on $[a, b]$. Therefore there exist positive real numbers $k_1, k_2$ such that $|f(x)|0$.
Since $f \in \mathcal{R}[a, b]$, there exists a partition $P_1$ of $[a, b]$ such that
$$U\left(P_1, f\right)-L\left(P_1, f\right)<\frac{\epsilon}{2}$$
Since $g \in \mathcal{R}[a, b]$, there exists a partition $P_2$ of $[a, b]$ such that
$$U\left(P_2, g\right)-L\left(P_2, g\right)<\frac{\epsilon}{2}$$
Let $P_0=P_1 \cup P_2$. Then $P_0$ is a refinement of $P_1$ as well as of $P_2$ and $L\left(P_1, f\right) \leq L\left(P_0, f\right) \leq U\left(P_0, f\right) \leq U\left(P_1, f\right)$;
$$L\left(P_2, g\right) \leq L\left(P_0, g\right) \leq U\left(P_0, g\right) \leq U\left(P_2, g\right)$$
So $U\left(P_0, f\right)-L\left(P_0, f\right) \leq U\left(P_1, f\right)-L\left(P_1, f\right)<\frac{\epsilon}{2}$ and $U\left(P_0, g\right)-L\left(P_0, g\right) \leq U\left(P_2, g\right)-L\left(P_2, g\right)<\frac{c}{2}$.
Let $P_0=\left(x_0, x_1, \ldots, x_n\right)$, where $a=x_0<x_1<\cdots<x_n=b$.
Let $M_r=\sup {x \in\left[x{r-1}, x_r\right]}(f+g)(x), m_r=\inf {x \in\left[x{r-1}, x_r\right]}(f+g)(x)$
\begin{aligned} & M_r^{\prime}=\sup {x \in\left[x{r-1}, x_r\right]} f(x), m_r^{\prime}=\inf {x \in\left[x{r-1}, x_r\right]} f(x) \ & M_r^{\prime \prime}=\sup {x \in\left[x{r-1}, x_r\right]} g(x), m_r^{\prime \prime}=\inf {x \in\left[x_r+1, x_r\right]} g(x), \text { for } r=1,2, \ldots, n . \end{aligned} Then $M_r \leq M_r^{\prime}+M_r^{\prime \prime}, m_r \geq m_r^{\prime}+m_r^{\prime \prime}$, for $r=1,2, \ldots, n$. $$\begin{array}{r} U\left(P_0, f+g\right)=M_1\left(x_1-x_0\right)+\cdots+M_n\left(x_n-x{n-1}\right) \ \leq\left[M_1^{\prime}\left(x_1-x_0\right)+\cdots+M_n^{\prime}\left(x_n-x_{n-1}\right)\right] \end{array}$$

\begin{aligned} & +\left[M_1^{\prime \prime}\left(x_1-x_0\right)+\cdots+M_n^{\prime \prime}\left(x_n-x_{n-1}\right)\right] \ & =U\left(P_0, f\right)+U\left(P_0, g\right) . \end{aligned}
Similarly, $L\left(P_0, f+g\right) \geq L\left(P_0, f\right)+L\left(P_0, g\right)$.
Hence $U\left(P_0, f+g\right)-L\left(P_0, f+g\right) \leq\left[U\left(P_0, f\right)-L\left(P_0, f\right)\right]+\left[U\left(P_0, g\right)-\right.$ $\left.L\left(P_0, g\right)\right]<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$.

## 数学代写|实分析代写Real Analysis代考|Some Riemann integrable functions

\text { We have } \begin{aligned} U(P, f)-L(P, f) & =\sum_{r=1}^n\left(M_r-m_r\right)\left(x_r-x_{r-1}\right) \ & =\sum_{r=1}^n\left{f\left(x_r\right)-f\left(x_{r-1}\right)\right}\left(x_r-x_{r-1}\right) \ & \leq|P| \sum_{r=1}^n\left{f\left(x_r\right)-f\left(x_{r-1}\right)\right} \ & =|P|{f(b)-f(a)}<\epsilon . \end{aligned}

$$U\left(P_1, f\right)-L\left(P_1, f\right)<\frac{\epsilon}{2}$$

$$U\left(P_2, g\right)-L\left(P_2, g\right)<\frac{\epsilon}{2}$$

$$L\left(P_2, g\right) \leq L\left(P_0, g\right) \leq U\left(P_0, g\right) \leq U\left(P_2, g\right)$$
$U\left(P_0, f\right)-L\left(P_0, f\right) \leq U\left(P_1, f\right)-L\left(P_1, f\right)<\frac{\epsilon}{2}$和$U\left(P_0, g\right)-L\left(P_0, g\right) \leq U\left(P_2, g\right)-L\left(P_2, g\right)<\frac{c}{2}$。

\begin{aligned} & M_r^{\prime}=\sup {x \in\left[x{r-1}, x_r\right]} f(x), m_r^{\prime}=\inf {x \in\left[x{r-1}, x_r\right]} f(x) \ & M_r^{\prime \prime}=\sup {x \in\left[x{r-1}, x_r\right]} g(x), m_r^{\prime \prime}=\inf {x \in\left[x_r+1, x_r\right]} g(x), \text { for } r=1,2, \ldots, n . \end{aligned}然后是$M_r \leq M_r^{\prime}+M_r^{\prime \prime}, m_r \geq m_r^{\prime}+m_r^{\prime \prime}$，对应$r=1,2, \ldots, n$。 $$\begin{array}{r} U\left(P_0, f+g\right)=M_1\left(x_1-x_0\right)+\cdots+M_n\left(x_n-x{n-1}\right) \ \leq\left[M_1^{\prime}\left(x_1-x_0\right)+\cdots+M_n^{\prime}\left(x_n-x_{n-1}\right)\right] \end{array}$$

\begin{aligned} & +\left[M_1^{\prime \prime}\left(x_1-x_0\right)+\cdots+M_n^{\prime \prime}\left(x_n-x_{n-1}\right)\right] \ & =U\left(P_0, f\right)+U\left(P_0, g\right) . \end{aligned}

## 数学代写|实分析代写Real Analysis代考|Worked Examples

\begin{aligned} & =\lim {x \rightarrow 0+} \frac{\cos x-1}{\sin x+x \cos x} \quad\left(=\frac{0}{0}\right) \ & =\lim {x \rightarrow 0+} \frac{-\sin x}{2 \cos x-x \sin x}=0 . \end{aligned}

$$=\lim {x \rightarrow 0^{+}} \frac{\frac{1}{x}}{-\frac{1}{x^2}}=\lim {x \rightarrow 0^{+}}(-x)=0 .$$

$\lim {x \rightarrow 0+} f(x)^{g(x)}$取不定式$1^{\infty}$。\begin{aligned} & \lim {x \rightarrow 0+} \log \left(\frac{\sin x}{x}\right)^{\frac{1}{x}}=\lim {x \rightarrow 0+} \frac{\log \frac{\sin x}{x}}{x} \quad\left(=\frac{0}{0}\right) \ &=\lim {x \rightarrow 0+} \frac{\frac{x}{\sin x} \cdot \frac{x \cos x-\sin x}{x^2}}{1} \ &=\lim {x \rightarrow 0+} \frac{x \cos x-\sin x}{x \sin x} \quad\left(=\frac{0}{0}\right) \ &=\lim {x \rightarrow 0+} \frac{-x \sin x}{x \cos x+\sin x}\left(=\frac{0}{0}\right) \ &=\lim {x \rightarrow 0+} \frac{-x \cos x-\sin x}{-x \sin x+2 \cos x}=0 . \end{aligned}因此$\lim {x \rightarrow 0+}\left(\frac{\sin x}{x}\right)^{\frac{1}{x}}=e^0=1$。

## 数学代写|实分析代写Real Analysis代考|Sign of the derivative

$f$ 据说在增加 $c$ 如果存在正数 $\delta$ 这样 $f(x)f(c)$ 对所有人 $x \in I$ 令人满意的 $cf(c)$ 对所有人 $x \in I$ 令人满意的 $c-\deltaf(c)$ 对所有人 $x \in I$ 令人满意的 $c<x<c+\delta$；
$f$ 据说在 $c$ 如果存在正数 $\delta$ 这样 $f(x)<f(c)$ 对所有人 $x \in I$ 令人满意的 $c<x<c+\delta$．

(i)如果$f^{\prime}(c)>0$，则$f$在$c$处增加
(ii)如果$f^{\prime}(c)<0$，则$f$在$c$处递减。证明。(i) $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}>0$。因此存在一个正的$\delta$，使得$\frac{f(x)-f(c)}{x-c}>0$对于所有的$x \in N^{\prime}(c, \delta) \cap I$。

## MATLAB代写

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