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