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# 数学代写|MAST20026 Real Analysis

## 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）

If in the field axioms for $\mathbb{R}$ we replace $\mathbb{R}$ by any other set with two operations + and – that satisfy these nine properties, then we say that that structure is a field. For example, $\mathbb{Q}$ is a field. The rules are valid since $\mathbb{Q} \subset \mathbb{R}$. The only thing that needs to be checked is that $a+b$ and $a \cdot b$ are in $\mathbb{Q}$ if both $a$ and $b$ are. For this reason $\mathbb{Q}$ is called a subfield of $\mathbb{R}$. Find another subfield.

Let $F$ be the set of all numbers of the form $x+y \sqrt{2}$ where $x, y \in \mathbb{Q}$. Again to be sure that nine properties of a field hold it is enough to check, here, that $a+b$ and $a \cdot b$ are in $F$ if both $a$ and $b$ are.

Using just the field axioms, show that
$$(x+1)^2=x^2+2 x+1$$
for all $x \in \mathbb{R}$. Would this identity be true in any field?

As a first step define what $x^2$ and $2 x$ really mean. In fact, define 2 . (It would be defined as $2=1+1$ since 1 and addition are defined in the field axioms.) Then multiply $(x+1) \cdot(x+1)$ using only the rules given here. Since your proof uses only the field axioms, it must be valid in any situation in which these axioms are true, not just for $\mathbb{R}$.

Using just the axioms, prove the arithmetic-geometric mean inequality:
$$\sqrt{a b} \leq \frac{a+b}{2}$$
for any $a, b \in \mathbb{R}$ with $a>0$ and $b>0$. (Assume, for the moment, the existence of square roots.)

Suppose $a>0$ and $b>0$ and $a \neq b$. Establish that $\sqrt{a} \neq \sqrt{b}$. Establish that
$$(\sqrt{a}-\sqrt{b})^2>0 \text {. }$$
Carry on. What have you proved? Now what if $a=b$ ?

Show for every nonempty, finite set $E$ that $\sup E=\max E$.

You can use induction on the size of $E$, that is, prove for every positive integer $n$ that if $E$ has $n$ elements, then
$$\sup E=\max E .$$

Without using the archimedean theorem, show that for each $x>0$ there is an $n \in \mathbb{N}$ so that $1 / n<x$.

Suppose not, then the set
$${1 / n: n=1,2,3, \ldots}$$
has a positive lower bound, etc. You will have to use the existence of a greatest lower bound.

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