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这是昆士兰大学 数学分析课程的代写成功案例。
MATH2400课程简介
Course Description: (Note: MATH1052 may be taken concurrently with MATH2400 by students who only took MATH1050 & MATH1051 in 1st year.) Bounded & monotone sequences. Sequences & series of real functions. Intermediate & mean value theorems, iterative procedures. Taylor’s Theorem & error estimates. Criteria for integrability. Vector functions, continuity & differentials. Implicit & Inverse Function Theorems & applications.
Assumed Background:
MATH1051, MATH1701, or have an equivalent qualification.
Prerequisites
The course is a concise and rigurous introduction to the central concept in mathematical analysis: limits. This concept is used to develop further notions such as continuity, the differential and integral calculus of functions of single and several variables, the implicit and inverse function theorems, uniform continuity of functions and the uniform convergence of sequences of functions. By the end of this course, you will have a solid understanding of the fundamental principles of mathematical analysis and be able to use them to solve mathematical problems, rather than just mechanically applying techniques without a true understanding of their significance.
There is some overlap between this course and MATH2401. MATH2401 goes into greater depth and contains more difficult proofs.
MATH2400 Mathematical Analysis HELP（EXAM HELP， ONLINE TUTOR）
(1) Find the supremum and infimum of the set
$$
S=\left{x \in \mathbb{R}: x<\frac{1}{x}\right} $$ Justify your answer. (2) Suppose $S$ is a non-empty subset of $\mathbb{R}$ which is bounded from above. Show that $\sup S=-\inf {-s: s \in S}$. (3) Show that if $A, B$ are bounded subsets of $\mathbb{R}$. Show that $\sup (A+B)=\sup A+\sup B, \quad$ and $\quad \inf (A+B)=\inf A+\inf B$ (4) Let $x>0$, show that there is $n \in \mathbb{N}$ such that $\frac{1}{2^n}0$, show that there exists a unique $m \in \mathbb{N}$ such that $m \leq x<m+1$. (In this case, we call such $m$ as $[x]$ ).
(1) If $b>a>0$, show that the sequence $x_n=\left(a^n+b^n\right)^{1 / n}$ is convergent with limit equal to $b$.
(2) If $\left{x_n\right}_{n=1}^{\infty}$ is a sequence of positive real number such that $\lim {n \rightarrow+\infty} x_n^{1 / n}=$ $L<1$, show that $x_n$ converges to 0 as $n \rightarrow+\infty$. (3) (a) Let $x_1=10$ and $x{n+1}=x_n+\frac{2}{x_n}$ for all $n \in \mathbb{N}$. Determine whether $\left{x_n\right}$ is convergent or not.
(b) Let $x_1=1$ and $x_{n+1}=\sqrt{4+x_n}$ for all $n \in \mathbb{N}$, show that $\left{x_n\right}$ is convergent. Find the limit.
(4) Show that if the sequence $\left{x_n\right}_{n=1}^{\infty}$ satisfies $x_n \geq 0$ for all $n$ and $\left{(-1)^n x_n\right}_{n=1}^{\infty}$ is convergent, then $\left{x_n\right}_{n=1}^{\infty}$ is convergent. Is the conclusion still true if we only assume $x_n \geq-1$ ?
(5) If $\left{x_n\right}_{n=1}^{\infty}$ is a sequence which is bounded from above. Let $s=\sup \left{x_n\right}$. Show that either $s=x_N$ for some $N$ or there is a subsequence $x_{n_k}$ so that $x_{n_k} \rightarrow s$ as $k \rightarrow+\infty$.
Q1. Let $a \in \mathbb{R}$ and $f:[0, a)$ be a real valued function given by $f(x)=x^4$.
(a) Show that $f$ is uniformly continuous.
(b) Is the conclusion in (a) still true if $a$ is replaced by $+\infty$ ? Justify your answer.
Q2. Let $f:[0,1] \rightarrow \mathbb{R}$ be a real valued function given by $f(x)=x^{1 / 3}$.
(a) Show that $f$ is not a Lipschitz function.
(b) Using $\varepsilon, \delta$ terminology, show that $f$ is uniformly continuous.
Q3. Let $f:[0,+\infty) \rightarrow \mathbb{R}$ be a continuous real valued function.
(a) Suppose there is $L, k>0$ such that for all $x>k,|f(x)| \leq$ L. Prove that $f$ is uniformly bounded by showing that there exists $\tilde{L}>0$ such that for all $x \in[0,+\infty),|f(x)| \leq \tilde{L}$.
(b) Suppose $\lim _{x \rightarrow+\infty} f(x)=\alpha \in \mathbb{R}$, show that $f$ is uniformly continuous on $[0,+\infty)$.
Q4. Let $f:[0,1] \rightarrow \mathbb{R}$ be a real valued function such that $f$ is continuous and $f(x) \notin \mathbb{Q}$ for all $x \in[0,1]$. Show that $f$ must be a constant function.
(a) Fix $\epsilon>0$. Since $\sum_{i=1}^{\infty} a_i$ converges, there exists $N \in \mathbb{N}$ such that for any $m, n>N$, $\sum_{i=m}^n a_i<\epsilon^{\frac{1}{3}}$. Then $\sum_{i=m}^n a_i^3<\left(\sum_{i=m}^n a_i\right)^3<\epsilon$. Hence, $\sum_{i=1}^{\infty} a_i^3$ converges.
(b) Counter-example: $\sum_{n=1}^{\infty} \frac{1}{n^3}$ converges but $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges. (See Textbook Example 3.7.6 for a proof of the divergence.)
(c) Since $a_n>0, b_n>\frac{a_1}{n}$. By Comparison Test, the divergence of $\sum_{i=n}^{\infty} b_n$ follows from that of $\sum_{n=1}^{\infty} \frac{1}{n}$.
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