1.1: The Real Number System

Having taken calculus, you know a lot about the real number system; however, you probably do not know that all its properties follow from a few basic ones. Although we will not carry out the development of the real number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also to focus on one property, completeness, that is probably new to you.

Field Properties

The real number system (which we will often call simply the reals) is first of all a set \(\{a, b, c, \cdots \}\) on which the operations of addition and multiplication are defined so that every pair of real numbers has a unique sum and product, both real numbers, with the following properties.

1. (A) a C b D b C a and ab D ba (commutative laws).
2. (B) .a C b/ C c D a C .b C c/ and .ab/c D a.bc/ (associative laws).
3. (C) a.b C c/ D ab C ac (distributive law).
4. (D) There are distinct real numbers 0 and 1 such that a C 0 D a and a1 D a for all a.
5. (E) For each a there is a real number a such that a C .a/ D 0, and if a ¤ 0, there is a real number 1=a such that a.1=a/ D

The manipulative properties of the real numbers, such as the relations

\[\begin{eqnarray*} (a+b)^2\ar=a^2+2ab+b^2,\\ (3a+2b)(4c+2d)\ar=12ac+6ad+8bc+4bd,\\ (-a)\ar=(-1)a,\quad a(-b)=(-a)b=-ab,\\ \arraytext{\phantom{xxx}and}\\ \frac{a}{b}+\frac{c}{d}\ar=\frac{ad+bc}{bd}\quad (b,d\ne0), \end{eqnarray*} \nonumber \]

all follow from –

. We assume that you are familiar with these properties.

A set on which two operations are defined so as to have properties –

is called a {}. The real number system is by no means the only field. The {} (which are the real numbers that can be written as \(r=p/q\), where \(p\) and \(q\) are integers and \(q\ne0\)) also form a field under addition and multiplication. The simplest possible field consists of two elements, which we denote by \(0\) and \(1\), with addition defined by \[\begin{equation}\label{eq:1.1.1} 0+0=1+1=0,\quad 1+0=0+1=1, \end{equation} \] and multiplication defined by \[\begin{equation}\label{eq:1.1.2} 0\cdot 0=0\cdot1=1\cdot 0=0,\quad 1\cdot1=1 \end{equation} \] (Exercise~).

The Order Relation

The real number system is ordered by the relation \(<\), which has the following properties.

A field with an order relation satisfying –

is an {} field. Thus, the real numbers form an ordered field. The rational numbers also form an ordered field, but it is impossible to define an order on the field with two elements defined by and so as to make it into an ordered field (Exercise~).

We assume that you are familiar with other standard notation connected with the order relation: thus, \(a>b\) means that \(b<a\); \(a\ge b\) means that either \(a=b\) or \(a>b\); \(a\le b\) means that either \(a=b\) or \(a<b\); the {} \(a\), denoted by \(|a|\), equals \(a\) if \(a\ge0\) or \(-a\) if \(a\le0\). (Sometimes we call \(|a|\) the {} of \(a\).)

You probably know the following theorem from calculus, but we include the proof for your convenience.

There are four possibilities:

Eq.~ holds in cases {} and {}, since \[\begin{equation} |a+b|= \begin{cases} |a|-|b|& \text{ if } |a| \ge |b|,\\ |b|-|a|& \text{ if } |b| \ge |a|. \end{cases} \tag*{\bbox} \end{equation} \]

The triangle inequality appears in various forms in many contexts. It is the most important inequality in mathematics. We will use it often.

Replacing \(a\) by \(a-b\) in yields \[ |a|\le|a-b|+|b|, \nonumber \] so \[\begin{equation} \label{eq:1.1.6} |a-b|\ge|a|-|b|. \end{equation} \] Interchanging \(a\) and \(b\) here yields \[ |b-a|\ge|b|-|a|, \nonumber \] which is equivalent to \[\begin{equation} \label{eq:1.1.7} |a-b|\ge|b|-|a|, \end{equation} \] since \(|b-a|=|a-b|\). Since \[ \big||a|-|b|\big|= \left\{\casespace\begin{array}{l} |a|-|b|\mbox{\quad if \quad} |a|>|b|,\\[2\jot] |b|-|a|\mbox{\quad if \quad} |b|>|a|, \end{array}\right. \nonumber \] and imply . Replacing \(b\) by \(-b\) in yields , since \(|-b|=|b|\).

With the real numbers associated in the usual way with the points on a line, these definitions can be interpreted geometrically as follows: \(b\) is an upper bound of \(S\) if no point of \(S\) is to the right of \(b\); \(\beta=\sup S\) if no point of \(S\) is to the right of \(\beta\), but there is at least one point of \(S\) to the right of any number less than \(\beta\) (Figure~).

This example shows that a supremum of a set may or may not be in the set, since \(S_1\) contains its supremum, but \(S\) does not.

A {} set is a set that has at least one member. The {}, denoted by \(\emptyset\), is the set that has no members. Although it may seem foolish to speak of such a set, we will see that it is a useful idea.

It is one thing to define an object and another to show that there really is an object that satisfies the definition. (For example, does it make sense to define the smallest positive real number?) This observation is particularly appropriate in connection with the definition of the supremum of a set. For example, the empty set is bounded above by every real number, so it has no supremum. (Think about this.) More importantly, we will see in Example~ that properties –

do not guarantee that every nonempty set that is bounded above has a supremum. Since this property is indispensable to the rigorous development of calculus, we take it as an axiom for the real numbers.

We first show that \(\beta=\sup S\) has properties and . Since \(\beta\) is an upper bound of \(S\), it must satisfy . Since any real number \(a\) less than \(\beta\) can be written as \(\beta-\epsilon\) with \(\epsilon=\beta-a>0\), is just another way of saying that no number less than \(\beta\) is an upper bound of \(S\). Hence, \(\beta=\sup S\) satisfies and

.

Now we show that there cannot be more than one real number with properties and . Suppose that \(\beta_1<\beta_2\) and \(\beta_2\) has property ; thus, if \(\epsilon>0\), there is an \(x_0\) in \(S\) such that \(x_0>\beta_2-\epsilon\). Then, by taking \(\epsilon=\beta_2-\beta_1\), we see that there is an \(x_0\) in \(S\) such that \[ x_0>\beta_2-(\beta_2-\beta_1)=\beta_1, \nonumber \] so \(\beta_1\) cannot have property . Therefore, there cannot be more than one real number that satisfies both and

.

We will often define a set \(S\) by writing \(S=\set{x}{\cdots}\), which means that \(S\) consists of all \(x\) that satisfy the conditions to the right of the vertical bar; thus, in Example~, \[\begin{equation}\label{eq:1.1.8} S=\set{x}{x<0} \end{equation} \] and \[ S_1=\set{x}{x\mbox{ is a negative integer}}. \nonumber \] We will sometimes abbreviate $x$ is a member of $S$'' by $x\in S$, and\(x\) is not a member of \(S\)’’ by \(x\notin S\). For example, if \(S\) is defined by , then \[ -1\in S\mbox{\quad but \quad} 0\notin S. \nonumber \]

Since \(n+1\) is an integer whenever \(n\) is, implies that \[ (n+1)\epsilon\le\beta \nonumber \] and therefore \[ n\epsilon\le\beta-\epsilon \nonumber \] for all integers \(n\). Hence, \(\beta-\epsilon\) is an upper bound of \(S\). Since \(\beta-\epsilon <\beta\), this contradicts the definition of~\(\beta\).

From Theorem~ with \(\rho=1\) and \(\epsilon=b-a\), there is a positive integer \(q\) such that \(q(b-a)>1\). There is also an integer \(j\) such that \(j>qa\). This is obvious if \(a\le0\), and it follows from Theorem~ with \(\epsilon=1\) and \(\rho=qa\) if \(a>0\). Let \(p\) be the smallest integer such that \(p>qa\). Then \(p-1\le qa\), so \[ qa<p\le qa+1. \nonumber \] Since \(1<q(b-a)\), this implies that \[ qa<p<qa+q(b-a)=qb, \nonumber \] so \(qa<p<qb\). Therefore, \(a<p/q<b\).

From Theorem~, there are rational numbers \(r_1\) and \(r_2\) such that \[\begin{equation} \label{eq:1.1.10} a<r_1<r_2<b. \end{equation} \] Let \[ t=r_1+\frac{1}{\sqrt2}(r_2-r_1). \nonumber \] Then \(t\) is irrational (why?) and \(r_1<t<r_2\), so \(a<t<b\), from .

A set \(S\) of real numbers is {} if there is a real number \(a\) such that \(x\ge a\) whenever \(x\in S\). In this case, \(a\) is a {} of \(S\). If \(a\) is a lower bound of \(S\), so is any smaller number, because of property

. If \(\alpha\) is a lower bound of \(S\), but no number greater than \(\alpha\) is, then \(\alpha\) is an {} of \(S\), and we write \[ \alpha=\inf S. \nonumber \] Geometrically, this means that there are no points of \(S\) to the left of \(\alpha\), but there is at least one point of \(S\) to the left of any number greater than \(\alpha\).

(Exercise~)

A set \(S\) is {} if there are numbers \(a\) and \(b\) such that \(a\le x\le b\) for all \(x\) in \(S\). A bounded nonempty set has a unique supremum and a unique infimum, and \[\begin{equation}\label{eq:1.1.11} \inf S\le\sup S \end{equation} \] (Exercise~).

A nonempty set \(S\) of real numbers is {} if it has no upper bound, or {} if it has no lower bound. It is convenient to adjoin to the real number system two fictitious points, \(+\infty\) (which we usually write more simply as \(\infty\)) and \(-\infty\), and to define the order relationships between them and any real number \(x\) by \[\begin{equation}\label{eq:1.1.12} -\infty<x<\infty. \end{equation} \] We call \(\infty\) and \(-\infty\) {}. If \(S\) is a nonempty set of reals, we write \[\begin{equation}\label{eq:1.1.13} \sup S=\infty \end{equation} \] to indicate that \(S\) is unbounded above, and \[\begin{equation}\label{eq:1.1.14} \inf S=-\infty \end{equation} \] to indicate that \(S\) is unbounded below.

The real number system with \(\infty\) and \(-\infty\) adjoined is called the {}, or simply the {}. A member of the extended reals differing from \(-\infty\) and \(\infty\) is {}; that is, an ordinary real number is finite. However, the word finite'' infinite real number’’ is redundant and used only for emphasis, since we would never refer to \(\infty\) or \(-\infty\) as real numbers.

The arithmetic relationships among \(\infty\), \(-\infty\), and the real numbers are defined as follows.

We also define \[ \infty+\infty=\infty\infty=(-\infty)(-\infty)=\infty \nonumber \] and \[ -\infty-\infty=\infty(-\infty)=(-\infty)\infty=-\infty. \nonumber \] Finally, we define \[ |\infty|=|-\infty|=\infty. \nonumber \]

The introduction of \(\infty\) and \(-\infty\), along with the arithmetic and order relationships defined above, leads to simplifications in the statements of theorems. For example, the inequality , first stated only for bounded sets, holds for any nonempty set \(S\) if it is interpreted properly in accordance with and the definitions of and . Exercises~ and

illustrate the convenience afforded by some of the arithmetic relationships with extended reals, and other examples will illustrate this further in subsequent sections.

It is not useful to define \(\infty-\infty\), \(0\cdot\infty\), \(\infty/\infty\), and \(0/0\). They are called {}, and left undefined. You probably studied indeterminate forms in calculus; we will look at them more carefully in Section~2.4.

These axioms are known as {}. The real numbers can be constructed from the natural numbers by definitions and arguments based on them. This is a formidable task that we will not undertake. We mention it to show how little you need to start with to construct the reals and, more important, to draw attention to postulate

, which is the basis for the principle of mathematical induction.

It can be shown that the positive integers form a subset of the reals that satisfies Peano’s postulates (with \(\overline{n}=1\) and \(n'=n+1\)), and it is customary to regard the positive integers and the natural numbers as identical. From this point of view, the principle of mathematical induction is basically a restatement of postulate

.

Let \[ \mathbb M=\set{n}{n\in \mathbb N\mbox{ and } P_n\mbox{ is true}}. \nonumber \] From , \(1\in \mathbb M\), and from , \(n+1\in \mathbb M\) whenever \(n\in \mathbb M\). Therefore, \(\mathbb M=\mathbb N\), by postulate

.

A proof based on Theorem~ is an {}, or {}. The assumption that \(P_n\) is true is the {}. (Theorem~ permits a kind of induction proof in which the induction assumption takes a different form.)

Induction, by definition, can be used only to verify results conjectured by other means. Thus, in Example~ we did not use induction to {} the sum \[\begin{equation}\label{eq:1.2.2} s_n=1+2+\cdots+n; \end{equation} \] rather, we {} that \[\begin{equation}\label{eq:1.2.3} s_n=\frac{n(n+1)}{2}. \end{equation} \] How you guess what to prove by induction depends on the problem and your approach to it. For example, might be conjectured after observing that \[ s_1=1=\frac{1\cdot2}{2},\quad s_2=3=\frac{2\cdot3}{2},\quad s_3=6=\frac{4\cdot3}{2}. \nonumber \] However, this requires sufficient insight to recognize that these results are of the form for \(n=1\), \(2\), and \(3\). Although it is easy to prove by induction once it has been conjectured, induction is not the most efficient way to find \(s_n\), which can be obtained quickly by rewriting as \[ s_n=n+(n-1)+\cdots+1 \nonumber \] and adding this to to obtain \[ 2s_n=[n+1]+\left[(n-1)+2\right]+\cdots+[1+n]. \nonumber \] There are \(n\) bracketed expressions on the right, and the terms in each add up to \(n+1\); hence, \[ 2s_n=n(n+1), \nonumber \] which yields .

The next two examples deal with problems for which induction is a natural and efficient method of solution.

The major effort in an induction proof (after \(P_1\), \(P_2\), , \(P_n\),
have been formulated) is usually directed toward showing that \(P_n\) implies \(P_{n+1}\). However, it is important to verify \(P_1\), since \(P_n\) may imply \(P_{n+1}\) even if some or all of the propositions \(P_1\), \(P_2\), , \(P_n\), are false.

The following formulation of the principle of mathematical induction permits us to start induction proofs with an arbitrary integer, rather than 1, as required in Theorem~.

For \(m\ge1\), let \(Q_m\) be the proposition defined by \(Q_m=P_{m+n_0-1}\). Then \(Q_1=P_{n_0}\) is true by . If \(m\ge1\) and \(Q_m=P_{m+n_0-1}\) is true, then \(Q_{m+1}=P_{m+n_0}\) is true by

with \(n\) replaced by \(m+n_0-1\). Therefore, \(Q_m\) is true for all \(m\ge1\) by Theorem~ with \(P\) replaced by \(Q\) and \(n\) replaced by \(m\). This is equivalent to the statement that \(P_n\) is true for all \(n\ge n_0\).

For \(n\ge n_0\), let \(Q_n\) be the proposition that \(P_{n_0}\), \(P_{n_0+1}\), , \(P_n\) are all true. Then \(Q_{n_0}\) is true by . Since \(Q_n\) implies \(P_{n+1}\) by

, and \(Q_{n+1}\) is true if \(Q_n\) and \(P_{n+1}\) are both true, Theorem~ implies that \(Q_n\) is true for all \(n\ge n_0\). Therefore, \(P_n\) is true for all \(n\ge n_0\).

One of our objectives is to develop rigorously the concepts of limit, continuity, differentiability, and integrability, which you have seen in calculus. To do this requires a better understanding of the real numbers than is provided in calculus. The purpose of this section is to develop this understanding. Since the utility of the concepts introduced here will not become apparent until we are well into the study of limits and continuity, you should reserve judgment on their value until they are applied. As this occurs, you should reread the applicable parts of this section. This applies especially to the concept of an open covering and to the Heine–Borel and Bolzano–Weierstrass theorems, which will seem mysterious at first.

We assume that you are familiar with the geometric interpretation of the real numbers as points on a line. We will not prove that this interpretation is legitimate, for two reasons: (1) the proof requires an excursion into the foundations of Euclidean geometry, which is not the purpose of this book; (2) although we will use geometric terminology and intuition in discussing the reals, we will base all proofs on properties –

(Section~1.1) and their consequences, not on geometric arguments.

Henceforth, we will use the terms {} and {} synonymously and denote both by the symbol \(\R\); also, we will often refer to a real number as a {} (on the real line).

In this section we are interested in sets of points on the real line; however, we will consider other kinds of sets in subsequent sections. The following definition applies to arbitrary sets, with the understanding that the members of all sets under consideration in any given context come from a specific collection of elements, called the {}. In this section the universal set is the real numbers.

Every set \(S\) contains the empty set \(\emptyset\), for to say that \(\emptyset\) is not contained in \(S\) is to say that some member of \(\emptyset\) is not in \(S\), which is absurd since \(\emptyset\) has no members. If \(S\) is any set, then \[ (S^c)^c=S\mbox{\quad and\quad} S\cap S^c=\emptyset. \nonumber \] If \(S\) is a set of real numbers, then \(S\cup S^c=\R\).

The definitions of union and intersection have generalizations: If \({\mathcal F}\) is an arbitrary collection of sets, then \(\cup\set{S}{S\in {\mathcal F}}\) is the set of all elements that are members of at least one of the sets in \({\mathcal F}\), and \(\cap\set{S}{S\in {\mathcal F}}\) is the set of all elements that are members of every set in \({\mathcal F}\). The union and intersection of finitely many sets \(S_1\), , \(S_n\) are also written as \(\bigcup^n_{k=1}S_k\) and \(\bigcap^n_{k=1}S_k\). The union and intersection of an infinite sequence \(\{S_k\}_{k=1}^\infty\) of sets are written as \(\bigcup^\infty_{k=1}S_k\) and \(\bigcap^\infty_{k=1}S_k\).

1pc 1pc

1pc If \(a\) and \(b\) are in the extended reals and \(a<b\), then the {} \((a,b)\) is defined by \[ (a,b)=\set{x}{a<x<b}. \nonumber \] The open intervals \((a,\infty)\) and \((-\infty,b)\) are {} if \(a\) and \(b\) are finite, and \((-\infty,\infty)\) is the entire real line.

The idea of neighborhood is fundamental and occurs in many other contexts, some of which we will see later in this book. Whatever the context, the idea is the same: some definition of closeness'' is given (for example, two real numbers areclose’‘if their difference is ``small’’), and a neighborhood of a point \(x_0\) is a set that contains all points sufficiently close to \(x_0\).

A {} of a point \(x_0\) is a set that contains every point of some neighborhood of \(x_0\) except for \(x_0\) itself. For example, \[ S=\set{x}{0<|x-x_0|<\epsilon} \nonumber \] is a deleted neighborhood of \(x_0\). We also say that it is a {} of \(x_0\).

Let \({\mathcal G}\) be a collection of open sets and \[ S=\cup\set{G}{G\in {\mathcal G}}. \nonumber \] If \(x_0\in S\), then \(x_0\in G_0\) for some \(G_0\) in \({\mathcal G}\), and since \(G_0\) is open, it contains some \(\epsilon\)-neighborhood of \(x_0\). Since \(G_0\subset S\), this \(\epsilon\)-neighborhood is in \(S\), which is consequently a neighborhood of \(x_0\). Thus, \(S\) is a neighborhood of each of its points, and therefore open, by definition.

Let \({\mathcal F}\) be a collection of closed sets and \(T =\cap\set{F}{F\in {\mathcal F}}\). Then \(T^c=\cup\set{F^c}{F\in {\mathcal F}}\) (Exercise~) and, since each \(F^c\) is open, \(T^c\) is open, from

. Therefore, \(T\) is closed, by definition.

Example~ shows that a set may be both open and closed, and Example~ shows that a set may be neither. Thus, open and closed are not opposites in this context, as they are in everyday speech.

It can be shown that the intersection of finitely many open sets is open, and that the union of finitely many closed sets is closed. However, the intersection of infinitely many open sets need not be open, and the union of infinitely many closed sets need not be closed (Exercises~ and ).

The next theorem says that \(S\) is closed if and only if \(S=\overline{S}\) (Exercise~).

Suppose that \(S\) is closed and \(x_0\in S^c\). Since \(S^c\) is open, there is a neighborhood of \(x_0\) that is contained in \(S^c\) and therefore contains no points of \(S\). Hence, \(x_0\) cannot be a limit point of \(S\). For the converse, if no point of \(S^c\) is a limit point of \(S\) then every point in \(S^c\) must have a neighborhood contained in \(S^c\). Therefore, \(S^c\) is open and \(S\) is closed.

Theorem~ is usually stated as follows.

Theorem~ and Corollary~ are equivalent. However, we stated the theorem as we did because students sometimes incorrectly conclude from the corollary that a closed set must have limit points. The corollary does not say this. If \(S\) has no limit points, then the set of limit points is empty and therefore contained in \(S\). Hence, a set with no limit points is closed according to the corollary, in agreement with Theorem~. For example, any finite set is closed. More generally, \(S\) is closed if there is a \(\delta>0\) such \(|x-y|\ge \delta\) for every pair \(\{x,y\}\) of distinct points in \(S\).

A collection \({\mathcal H}\) of open sets is an {} of a set \(S\) if every point in \(S\) is contained in a set \(H\) belonging to \({\mathcal H}\); that is, if \(S\subset\cup\set{H}{H\in {\mathcal H}}\).

Since \(S\) is bounded, it has an infimum \(\alpha\) and a supremum \(\beta\), and, since \(S\) is closed, \(\alpha\) and \(\beta\) belong to \(S\) (Exercise~). Define \[ S_t=S\cap [\alpha,t] \mbox{\quad for \ } t\ge\alpha, \nonumber \] and let \[ F=\set{t}{\alpha\le t\le\beta and finitely many sets from \mathcal H cover S_t}. \nonumber \] Since \(S_\beta=S\), the theorem will be proved if we can show that \(\beta \in F\). To do this, we use the completeness of the reals.

Since \(\alpha\in S\), \(S_\alpha\) is the singleton set \(\{\alpha\}\), which is contained in some open set \(H_\alpha\) from \({\mathcal H}\) because \({\mathcal H}\) covers \(S\); therefore, \(\alpha\in F\). Since \(F\) is nonempty and bounded above by \(\beta\), it has a supremum \(\gamma\). First, we wish to show that \(\gamma=\beta\). Since \(\gamma\le\beta\) by definition of \(F\), it suffices to rule out the possibility that \(\gamma<\beta\). We consider two cases.

{Case 1}. Suppose that \(\gamma<\beta\) and \(\gamma\not\in S\). Then, since \(S\) is closed, \(\gamma\) is not a limit point of \(S\) (Theorem~). Consequently, there is an \(\epsilon>0\) such that \[ [\gamma-\epsilon,\gamma+\epsilon]\cap S=\emptyset, \nonumber \] so \(S_{\gamma-\epsilon}=S_{\gamma+\epsilon}\). However, the definition of \(\gamma\) implies that \(S_{\gamma-\epsilon}\) has a finite subcovering from \({\mathcal H}\), while \(S_{\gamma+\epsilon}\) does not. This is a contradiction.

{Case 2}. Suppose that \(\gamma<\beta\) and \(\gamma\in S\). Then there is an open set \(H_\gamma\) in \({\mathcal H}\) that contains \(\gamma\) and, along with \(\gamma\), an interval \([\gamma-\epsilon,\gamma+\epsilon]\) for some positive \(\epsilon\). Since \(S_{\gamma-\epsilon}\) has a finite covering \(\{H_1, \dots,H_n\}\) of sets from \({\mathcal H}\), it follows that \(S_{\gamma+\epsilon}\) has the finite covering \(\{H_1, \dots,H_n,H_\gamma\}\). This contradicts the definition of \(\gamma\).

Now we know that \(\gamma=\beta\), which is in \(S\). Therefore, there is an open set \(H_\beta\) in \({\mathcal H}\) that contains \(\beta\) and along with \(\beta\), an interval of the form \([\beta-\epsilon,\beta+\epsilon]\), for some positive \(\epsilon\). Since \(S_{\beta-\epsilon}\) is covered by a finite collection of sets \(\{H_1, \dots,H_k\}\), \(S_\beta\) is covered by the finite collection \(\{H_1, \dots, H_k, H_\beta\}\). Since \(S_\beta=S\), we are finished.

Henceforth, we will say that a closed and bounded set is {}. The Heine–Borel theorem says that any open covering of a compact set \(S\) contains a finite collection that also covers \(S\). This theorem and its converse (Exercise~) show that we could just as well define a set \(S\) of reals to be compact if it has the Heine–Borel property; that is, if every open covering of \(S\) contains a finite subcovering. The same is true of \(\R^n\), which we study in Section~5.1. This definition generalizes to more abstract spaces (called {}) for which the concept of boundedness need not be defined.

As an application of the Heine–Borel theorem, we prove the following theorem of Bolzano and Weierstrass.

We will show that a bounded nonempty set without a limit point can contain only a finite number of points. If \(S\) has no limit points, then \(S\) is closed (Theorem~) and every point \(x\) of \(S\) has an open neighborhood \(N_x\) that contains no point of \(S\) other than \(x\). The collection \[ {\mathcal H}=\set{N_x}{x\in S} \nonumber \] is an open covering for \(S\). Since \(S\) is also bounded, Theorem~ implies that \(S\) can be covered by a finite collection of sets from \({\mathcal H}\), say \(N_{x_1}\), , \(N_{x_n}\). Since these sets contain only \(x_1\), , \(x_n\) from \(S\), it follows that \(S=\{x_1, \dots,x_n\}\).