MY-ASSIGNMENTEXPERT™可以为您提供 handbook MAST20026 Real Analysis实分析的代写代考和辅导服务！
这是墨尔本大学 实分析课程的代写成功案例。
MAST20026课程简介
This subject introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. Many of the important results are proved rigorously and students are introduced to methods of proof such as mathematical induction and proof by contradiction.
The important distinction between the real numbers and the rational numbers is emphasized and used to motivate rigorous notions of convergence and divergence of sequences, including the Cauchy criterion. These ideas are extended to cover the theory of infinite series, including common tests for convergence and divergence. A similar treatment of continuity and differentiability of functions of a single variable leads to applications such as the Mean Value Theorem and Taylor’s theorem. The definitions and properties of the Riemann integral allow rigorous proof of the Fundamental Theorem of Calculus. The convergence properties of sequences and series are explored, with applications to power series representations of elementary functions and their generation by Taylor series. Fourier series are introduced as a way to represent periodic functions.
Prerequisites
On completion of this subject students should
- Acquire an appreciation of rigour in mathematics, be able to use proof by induction, proof by contradiction, and to use epsilon-delta proofs both as a theoretical tool and a tool of approximation;
- Understand the theory and applications of the Riemann integral and improper integrals;
- Be able to determine the convergence and divergence of infinite series;
- Have a good knowledge of the theory and practice of power series expansions and Taylor polynomial approximations; and
- Understand the role of Fourier series in representing periodic functions.
MAST20026 Real Analysis HELP（EXAM HELP， ONLINE TUTOR）
A nonempty subset $X \subset \mathbb{R}$ is called perfect if it has no isolated points or equivalently if it is closed and each open interval containing any point in $X$ contains infinitely many points of $X$). Show that the Cantor set is perfect. Hint: The endpoints of all of the subintervals occurring in the Cantor construction belong to C.
Prove that every perfect set $X \subset \mathbb{R}$ is uncountable. Hint: If $X$ is countable, construct a descending sequence of bounded, closed subsets of $X$ whose intersection is empty.
A subset $A$ of $\mathbb{R}$ is said to be nowhere dense in $\mathbb{R}$ provided that every open set $U$ has an open subset that is disjoint from $A$. Show that the Cantor set is nowhere dense in $\mathbb{R}$.
Let $f$ be a continuous function and $B$ be a Borel set. Show that $f^{-1}(B)$ is a Borel set. Hint: The collection of sets $E$ for which $f^{-1}(E)$ is Borel is a $\sigma$-algebra containing the open sets.
Use the preceding two problems to show that a continuous strictly increasing function that is defined on an interval maps Borel sets to Borel sets.
MY-ASSIGNMENTEXPERT™可以为您提供 HANDBOOK MAST20026 REAL ANALYSIS实分析的代写代考和辅导服务！
