12 1. From the Real Numbers to the Complex Numbers

When there is no real number L for which {xn} converges to L, we say that

{xn} diverges.

The deﬁnition of the limit demands that the terms eventually get arbitrarily

close to a given L. The deﬁnition of a Cauchy sequence states that the terms of

the sequence eventually get arbitrarily close to each other. The most fundamental

result in real analysis is that a sequence of real numbers converges if and only if it

is a Cauchy sequence. The word complete has several similar uses in mathematics;

it often refers to a metric space in which being Cauchy is a necessary and suﬃcient

condition for convergence of a sequence. See Section 6. The following subtle remark

indicates a slightly diﬀerent way one can deﬁne the real numbers.

Remark 3.1. Consider an ordered ﬁeld F satisfying the Archimedean property.

In other words, given positive elements x and y, there is an integer n such that y

added to itself n times exceeds x. Of course we write ny for this sum. It is possible

to consider limits and Cauchy sequences in F. Suppose that each Cauchy sequence

in F has a limit in F. One can then derive the least upper bound property, and F

must be the real numbers R. Hence we could give the deﬁnition of the real number

system by decreeing that R is an ordered ﬁeld satisfying the Archimedean property

and that R is complete in the sense of Cauchy sequences.

We return to the real numbers. A sequence {xn} of real numbers is bounded

if and only if its set of values is a bounded subset of R. A convergent sequence

must of course be bounded; with ﬁnitely many exceptions all the terms are within

distance 1 from the limit. Similarly a Cauchy sequence must be bounded; with

ﬁnitely many exceptions all the terms are within distance 1 of some particular xN .

Proving that a convergent sequence must be Cauchy uses what is called an

2

argument. Here is the idea: if the terms are eventually within distance

2

of

some limit L, then they are eventually within distance of each other. Proving the

converse assertion is much more subtle; somehow one must ﬁnd a candidate for the

limit just knowing that the terms are close to each other. See for example [8,20].

The proofs rely on the notion of subsequence, which we deﬁne now, but which we

do not use meaningfully until Chapter 8. Let {xn} be a sequence of real numbers

and let k → nk be an increasing function. We write {xnk } for the subsequence of

{xn} whose k-th term is xnk . The proof that a Cauchy sequence converges amounts

to ﬁrst ﬁnding a convergent subsequence and then showing that the sequence itself

converges to the same limit.

We next prove a basic fact that often allows us to determine convergence of a

sequence without knowing the limit in advance. A sequence {xn} is called nonde-

creasing if, for each n, we have xn+1 ≥ xn. It is called nonincreasing if, for each

n, we have xn+1 ≤ xn. It is called monotone if it is either nonincreasing or nonde-

creasing. The following fundamental result, illustrated by Figure 1.4, will get used

occasionally in this book. It can be used also to establish that a Cauchy sequence

of real numbers has a limit.

Proposition 3.3. A bounded monotone sequence of real numbers has a limit.

Proof. We claim that a nondecreasing sequence converges to its least upper bound

(supremum) and that a nonincreasing sequence converges to its greatest lower