$2.10$ Completeness of $\mathbb{R}$
By Theorem $2.7 .9$, between any two real numbers, points on the number line, there exists a rational number. This suggests there are a lot of rational numbers, but it is not clear from this Theorem whether the entire real line consists of only rational numbers. Some people might wish this were the case because then each real number could be described, not just as a point on a line but also algebraically, as the quotient of integers. Before 500 B.C., a group of mathematicians, led by Pythagoras believed in this, but they discovered their beliefs were false. It happened roughly like this. They knew they could construct the square root of two as the diagonal of a right triangle in which the two sides have unit length; thus they could regard $\sqrt{2}$ as a number. Unfortunately, they were also sble to show $\sqrt{2}$ could not be written as the quotient of two integers. This discovery that the rational numbers could not even account for the results of geometric constructions was very upsetting to the Pythagoreans, especially when it became clear there were an endless supply of such “irrational” numbers.
This shows that if it is desired to consider all points on the number line, it is necessary to abandon the attempt to describe arbitrary real numbers in a purely algebraic manner using only the integers. Some might desire to throw out all the irrational numbers, and considering only the rational numbers, confine their attention to algebra, but this is not the approsch to be followed here because it will effectively eliminate every major theorem of calculus and analysis. In this book real numbers will continue to be the points on the number line, a line which has no holes. This lack of holes is more precisely described in the following way.
Definition 2.10.1 A non empty set, $S \subseteq \mathbb{R}$ is bounded above (below) if there exists $x \in \mathbb{R}$ such that $x \geq(\leq) s$ for all $s \in S$. If $S$ is a nonempty set in $\mathbb{R}$ which is bounded above, then a number, l which has the property that l is an upper bound and that every other upper bound is no smaller than $l$ is called a least upper bound, l.u.b. (S) or often sup $(S)$. If $S$ is a nonempty set bounded below, define the greatest lower bound, g.l.b. $(S)$ or $\inf (S)$ similarly. Thus $g$ is the g.l.b. $(S)$ means $g$ is a lower bound for $S$ and it is the largest of all lower bounds. If $S$ is a nonempty subset of $\mathbb{R}$ which is not bounded above, this information is expressed by saying sup $(S)=+\infty$ and if $S$ is not bounded below, $\inf (S)=-\infty$.
In an appendix, there is a proof that the real numers can be obtained as equivalence classes of Cauchy sequences of rational numbers but in this book, we follow the historical development of the subject and accept it as an axiom. In other words, we will believe in the real numbers and this axiom.
The completeness axiom was identified by Bolzano as the reason for the truth of the intermediate value theorem for continuous functions around 1818. However, every existence theorem in calculus depends on some form of the completeness axiom.
Axiom 2.10.2 (completeness) Every nonempty set of real numbers which is bounded above has a least upper bound and every nonempty set of real numbers which is bounded below has a greatest lower bound.
It is this axiom which distinguishes Calculus from Algebra. A fundamental result shout sup and inf is the following.
Proposition 2.10.3 Let $S$ be a nonempty set and suppose sup $(S)$ exists. Then for every $\delta>0$,
$$
S \cap(\sup (S)-\delta, \sup (S)] \neq \emptyset
$$
If $\inf (S)$ exists, then for every $\delta>0$,
$$
S \cap[\inf (S), \inf (S)+\delta) \neq \varphi .
$$

$2.10$ $\mathbb{R}$ 的完整性由定理$2.7 .9$，在任意两个实数之间，数轴上的点，存在一个有理数。这表明有很多有理数，但从这个定理不清楚整条实线是否仅由有理数组成。有些人可能希望是这样，因为这样每个实数都可以被描述为，不仅是一条线上的一个点，而且还可以代数地描述为整数的商。公元前 500 年之前，一群以毕达哥拉斯为首的数学家相信这一点，但他们发现他们的信念是错误的。大致是这样发生的。他们知道他们可以将 2 的平方根构造为直角三角形的对角线，其中两条边都有单位长度。因此他们可以将 $\sqrt{2}$ 视为一个数字。不幸的是，他们也很容易证明 $\sqrt{2}$ 不能写成两个整数的商。有理数甚至不能解释几何构造的结果这一发现让毕达哥拉斯学派非常沮丧，尤其是当发现这种“无理数”源源不断地供应时尤其如此。这表明，如果希望考虑数轴上的所有点，则必须放弃仅使用整数以纯代数方式描述任意实数的尝试。有些人可能希望抛弃所有无理数，只考虑有理数，将注意力集中在代数上，但这不是这里要遵循的方法，因为它会有效地消除微积分和分析的每一个主要定理。在本书中，实数将继续是数轴上的点，一条没有孔的线。用以下方式更准确地描述了这种缺乏孔。定义 2.10.1 一个非空集，如果存在 $x \in \mathbb{R}$ 使得 $x \geq(\leq) s$，则 $S \subseteq \mathbb{R}$ 有界上（下）对于所有 $s \in S$。如果 $S$ 是 $\mathbb{R}$ 中的一个非空集合，它是有界的，那么一个数 l 具有以下性质： l 是一个上界并且每隔一个上界不小于 $l$ 是称为最小上限，lub (S) 或经常吃 $(S)$。如果 $S$ 是下界的非空集，则定义最大下界 g.l.b。 $(S)$ 或 $\inf (S)$ 类似。因此 $g$ 是 g.l.b。 $(S)$ 意味着 $g$ 是 $S$ 的下限，它是所有下限中最大的。如果 $S$ 是 $\mathbb{R}$ 的一个非空子集，它没有上界，则此信息表示为 sup $(S)=+\infty$ 并且如果 $S$ 没有下界，则 $\ inf (S)=-\infty$。在附录中，有一个证明，实数可以作为有理数的柯西序列的等价类获得，但在本书中，我们遵循该主题的历史发展并将其作为公理接受。换句话说，我们将相信实数和这个公理。完备性公理在 1818 年左右被 Bolzano 确定为连续函数的中间值定理为真的原因。然而，微积分中的每个存在定理都依赖于某种形式的完备性公理。公理 2.10.2（完备性）上界的每个非空实数集都有一个最小上界，而下界的每个非空实数集都有一个最大下界。正是这个公理将微积分与代数区分开来。一个基本的结果喊 sup 和 inf 如下。命题 2.10.3 令 $S$ 是一个非空集合并假设 sup $(S)$ 存在。那么对于每一个$\delta>0$， $$ S \cap(\sup (S)-\delta, \sup (S)] \neq \emptyset $$ 如果 $\inf (S)$ 存在，那么对于每个 $\delta>0$， $$ S \cap[\inf (S), \inf (S)+\delta) \neq \varphi 。 $$