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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）

In Section 1.2 we made much of the fact that there is a number whose square is 2 and so $\sqrt{2}$ does exist as a real number. Show that
$$\alpha=\sup \left{x \in \mathbb{R}: x^2<2\right}$$
exists as a real number and that $\alpha^2=2$.

Not that easy to show. Rule out the possibilities $\alpha^2<2$ and $\alpha^2>2$ using the archimedean property to assist.

Show that the numbers of the form
$$\pm m \sqrt{2} / n$$
for $m, n \in \mathbb{N}$ are dense.

To find a number in $(x, y)$, find a rational in $(x / / \sqrt{2}, y / / \sqrt{2})$. Conclude from this that the set of all (irrational) numbers of the form $\pm m \sqrt{2} / n$ is dense.

Let $G$ be a subgroup of the real numbers under addition (i.e., if $x$ and $y$ are in $G$, then $x+y \in G$ and $-x \in G$ ). Show that either $G$ is a dense subset of $\mathbb{R}$ or else there is a real number $\alpha$ so that
$$G={n \alpha: n=1, \pm 1, \pm 2, \pm 3, \ldots} .$$

If $G={0}$, then take $\alpha=0$. If not, let $\alpha=\inf G \cap(0, \infty)$. Case 1: If $\alpha=0$ show that $G$ is dense. Case 2: If $\alpha>0$ show that
$$G={n \alpha: n=1, \pm 1, \pm 2, \pm 3, \ldots} .$$
For case 1 consider an interval $(r, s)$ with $r<s$. We wish to find a member of $G$ in that interval. To keep the argument simple just consider, for the moment, the situation in which $0<r<s$. Choose $g \in G$ with $0<g<s-r$. The set
$$M={n \in \mathbb{N}: n g \geq s}$$
is nonempty (why?) and so there is a minimal element $m$ in $M$ (why?). Now check that $(m-1) g$ is in $G$ and inside the interval $(r, s)$.

On IQ tests one frequently encounters statements such as “what is the next term in the sequence $3,1,4,1,5, \ldots$ ?”. In terms of our definition of a sequence is this correct usage? (By the way, what do you suppose the next term in the sequence might be?)

For the next term in the sequence some people might expect a 1 . Most mathematicians would expect a 9 .

Give two different formulas (for two different sequences) that generate a sequence whose first four terms are $2,4,6,8$.

Here is a formula that generates the first five terms of the sequence $0,0,0,0, c$,
$$f(n)=\frac{c(n-1)(n-2)(n-3)(n-4)}{4 !}$$

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