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the definitions of a continuous function, the derivative of a function, and even the

number e.

Our study of nearness begins with a careful look at sequences and their limits:

first in RвЂ™ in Section 12.2, and then in RвЂќ in Section 12.3. Sections 12.4 and

12.5 define open and closed sets and describe the complementarity between these

two topics. Open and closed sets play an important role in clarifying the hypothe-

ses behind most economic principles. For example, theorems which characterize

economic equilibria often require that the underlying space of commodities be a

closed and bounded set since the existence of an equilibrium is guaranteed for

such sets. Section 12.6 discusses the properties of closed and bounded sets.

The exposition of this chapter follows a careful logical development as one

principle is deduced from previous ones. As a result, this chapter has more proofs

in it than earlier chapters. Most of these proofs arc short and straightforward. Work

at understanding these proofs since the ability to follow a logical argument and

occasionally even to produce one is a valuable asset for working effectively in

economics.

12.1 SEQUENCES OF REAL NUMBERS

The natural numbers-also called the positive integers

˜ are just the usual

counting numbers: 1, 2,3,4,. A sequence of real numbers is an assignment

of a real number to each natural number. A sequence is usually written as

254 LIMITS AND OPEN SETS I1 21

1 xl,

x2, x3,. , x,,, .}, where xi is the real number assigned to the natural number

1, thefirst number in the sequence; x2 is the real number assigned to 2, the second

number in the sequence; and so on.

Some examples of a sequence of real numbers are:

Example 12.1

I

I h) {1,4, 1,5,9, ,,,}.

g) {3.1,3.14,3.141,3.1415 ,_.. },

For each natural number n, each sequence has a well-defined nth number x,,. For

example, the nth number in sequence b is l/n, the nth number in sequence e is

(-l)вЂњ, the nth number in sequence f is (n + 1)/n, and the nth number in sequence

g is the truncation of the decimal expansion of the number r to n decimal places.

We sometimes write a typical sequence {x1, x2, ,x3,. .} as {x.},=,

Limit of a Sequence

There are basically three kinds of sequences:

(I) sequences like b, f, and g. in which the entries get closer and closer and

stay close to some limiting value;

(2) sequences like a in which the entries increase without bound; and

(3) sequences like c, d, E, and h in which neither behavior occurs so that the

entries jump back and forth on the number line.

We are most interested in the first type of sequence-the one in which the entries

approach arbitrarily close and stay arbitrarily close to some real number, called

the limir of the sequence. Notice that we need both parts of this statement. The

entries of sequence c get arbitrarily close to 0 but they do not stay there; the entries

of sequence d get arbitrarily close to both + I and -1, but the sequence does not

stay close to either. Neither sequence c nor sequenced has a definite limit.

In order to define the concept of a limit carefully, we need to formalize the

notion that number s is close to number r ifs lies in some small interval about

r. More precisely, let E (epsilon) denote a small, positive real number, as is the

custom in mathematics. Then, the &-interval about the number r is defined to be

the interval

I,(r) = {s E R : 1s - rl < E}. (1)

255

[lZ.l] IEQUEKES OF REAL NUMBtRS

In interval notation, I,(r) = (I - 6 r + 8). Intuitively speaking, ifs is in I,(r) and

if 8 is small, then s is вЂњcloseвЂќ to r. The smaller E is, the closer s is to r.

Definition Let {xl, x2, x3, .} be a sequence of real numbers and let I be a real

number. We say that r is the limit of this sequence if for an)? (small) positive

number E, there is a positive integer N such that for all II 5 N, x, is in the

&-interval about r; that is,

In this case, we say that the sequence converges to I and we write

limx, = I or limx = r or simply x,, - r.

,/_I вЂ˜1

This definition states that x,, converges to I if no matter how small an interval

one chooses about I, from some point on (n 2 N in the above definition) the

entries of the sequence get into and stay in that interval. Of course, how scan the

sequence gets into the interval will? in general, depend on the size of the interval;

in other words. N depends on the size of F. In the above examples, sequence h

converges to 0, sequence f converges to 1, and sequence g convqes to n.

Example 12.2 Here are three mwe sequences which converge to 0:

Notice that the elements of the converging sequence need not be distinct

from each other or distinct from the limit, as the first sequence in Example 12.2

illustntcs. The convergence need not be all from one side, down to the limit, or up

to the limit, as the second sequence illustrates. Finally, the convergence need not be

moontonic: each element need not bc closer to the limit than all previous elements.

This is illustrated by the third sequence in Example 12.2. All that convergence

requires is that ultimately the elrmcnts remain within any prespecified distance of

the limit.

As mentioned above, sequences c, d. and e in Example 12.1 do not converge

to a limit. Although their entries get arbitrarily close to some numberPC in c;

and + 1 and I in d and e-they donвЂ˜t stay close. If the entries of a sequence

get arbitrarily close to some number. that number is called an accumulation point

of the sequence. More formally. r is an accumulation point or cluster point

of the sequence {x.} if for any positive E there are infinifely many elements of

the sequence in the interval I,(r), as defined in (1). A limit is a special case of

an accumulation point. A sequence can have a number of different accumulation

points, as illustrated by sequences d and e in Example 12.1. However, a sequence

can have only one limit.

Theorem 12.1 A sequence can have at most one limit. I

We want to formalize the intuitive notion that a sequence cannot get

roof

arbitrarily close and sray arbitrarily close to two different points. Suppose that a

sequence {x.}P;=, has two limits: r, and r2. Take E to be some number less than

half the distance between rl and rz, say E = i!rl - ˜1, so that I&r,) and I,(Q)

are disjoint intervals, as in Figure 12.1, Since x, - rlr there is an N1 such that

for n 2 NI all the x. are in I,(rl); and since n, - 12, there is an Nz such that

for all n 2 Nz all the xa are in IE(r2). Therefore, for u/l n 2 max{N,, Nz}, xвЂќ

are in both I=(r,) and Ie(rz). But no point can be in both Ie(rI) and /,(˜-a

contradiction which proves the theorem. n

I&, I,($

?A˜˜ _

I--,

r1-˜- , вЂ˜1 вЂ˜I, 2

Figure

The intervals Ic(rl) and l,(r?) are disjoinr

12.1 for E = $ lr, r21

In studying sequences, one often must consider the subsequences of a given

sequence. To define the concept of a subsequence carefully. think of a sequence as

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