## 数学代写|实分析作业代写Real analysis代考|Rational numbers

The most basic numbers, namely, the natural numbers, are defined axiomatically via set theory. 7.ero and the negative integers are added via algebraic axioms and properties. We take all this as given because they are not analytic in nature. We use them to define fractions and establish their algebraic properties. Taking fractions,or rational numbers, as a starting point may seem arbitrary, but greater exposure to these numbers serves a useful pedagogical purpose since rational numbers play a critical role in proving many results in real analysis.

The ordinary addition and multiplication of rational numbers is defined the way we remember them from past algebra experience:
$$\frac{j}{k}+\frac{m}{n}=\frac{k m+j n}{k n}, \quad \frac{j}{k} \frac{m}{n}=\frac{j m}{k n}$$
Here $j, m \in \mathbb{Z}$, and $k, n \in \mathbb{N}$. Since sums and products of integers are again integers, the above definitions indicate that sums and products of rational numbers are also rational.

When adding rational numbers, we often use the least common multiple of the denominators $k$ and $n$ to speed up the process; if not, then the greatest common divisor of the two denominators is canceled out in the reduced form as in
$$\frac{19}{18}-\frac{11}{12}=\frac{6 \times 2 \times 19-6 \times 3 \times 11}{6 \times 3 \times 6 \times 2}=\frac{6(38-33)}{6(36)}=\frac{5}{36}$$
Notice that the least common multiple of the denominators 12 and 18 is indeed 36.

The set $\mathbb{Q}$ is a more structured set than $\mathbb{N}$ or $\mathbb{Z}$. From an algebraic point of view, $\mathbb{Q}$ is a field, but neither $\mathbb{N}$ nor $\mathbb{Z}$ are fields. Specifically, when we divide two numbers in $\mathbb{Q}$, the result is in $\mathbb{Q}$ (as long as we do not divide by 0 ) because if $p=m / n$ and $q=j / k$ then
$$\frac{p}{q}=p \frac{1}{q}=\frac{m}{n} \frac{k}{j}=\frac{m k}{n j}$$
is again rational; for instance, operations as simple as taking the average of two numbers $(p+q) / 2$ are possible in $\mathbb{Q}$ but not in $\mathbb{Z}$

## 数学代写|实分析作业代写Real analysis代考|Cauchy sequences

Our goal is to start from the set $\mathbb{Q}$ of all rational numbers and build up to the set of all real numbers $\mathbb{R}$, including all of the irrationals. We now examine sequences of rational numbers that we can be sure will converge without requiring a knowledge of their limits. This suits our purpose since we don’t know all the irrational numbers; but is it possible?

We must rule out rational sequences that don’t converge, like the sequences $q_n=$ $n^2\left(\right.$ or $\left.1^2, 2^2, 3^2, \ldots\right)$ and $q_n=(-1)^n$ (or $\left.-1,1,-1,1, \ldots\right)$, which don’t approach any number, be it rational or not. So what is different about the sequence in (4.1) that does converge? How can we tell if a sequence converges if we don’t know a limit for it?

Our next major step on the way to the real numbers is to identify these special sequences of rational numbers.

Earlier we saw how to estimate $\sqrt{2}$ using a sequence of rational numbers. We found that the more terms we used, the closer the approximation became. Now since we don’t know the exact digital value of $\sqrt{2}$, consider a practical question: how many terms of a rational sequence should we use to get a desired accuracy, say, 10 decimal places?

If we use many terms of the sequence, then the later terms form a dense patch of rationals around the limit; in the language of Chapter 3 , the numbers cluster around the limit. The distance between any pair of terms $q_m$ and $q_n$ with large enough indices $m$ and $n$ can be as small as we need it to be, like a target threshold for a desired level of accuracy. For the 10-decimal-place accuracy, this threshold is $0.00000000005$ or $5 \times 10^{-11}$.

More precisely, we must reach an index $N$ that is large enough that the difference between any pair of terms $q_m$ and $q_n$ is less than $0.00000000005$ as long as $m, n$ both exceed $N$. Let’s illustrate this idea using the recursion (1.3) in Exercise 3 (the divide and average rule).

# 实分析代考

## 数学代写|实分析作业代写Real analysis代考|Rational numbers

$$\frac{j}{k}+\frac{m}{n}=\frac{k m+j n}{k n}, \quad \frac{j}{k} \frac{m}{n}=\frac{j m}{k n}$$

$$\frac{19}{18}-\frac{11}{12}=\frac{6 \times 2 \times 19-6 \times 3 \times 11}{6 \times 3 \times 6 \times 2}=\frac{6(38-33)}{6(36)}=\frac{5}{36}$$

$$\frac{p}{q}=p \frac{1}{q}=\frac{m}{n} \frac{k}{j}=\frac{m k}{n j}$$

## 数学代写|实分析作业代写Real analysis代考|Cauchy sequences

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: