26 I The Expanding Universe of Numbers
Proposition 25Any fundamental sequence of real numbers is convergent.
Proof If{an}is a fundamental sequence of real numbers, then{an}is bounded and,
for anyε>0, there exists a positive integerm=m(ε)such that
−ε/ 2 <ap−aq<ε/2forallp,q≥m.But, by Proposition 24, the sequence{an}has a convergent subsequence{ank}.Iflis
the limit of this subsequence, then there exists a positive integerN≥msuch that
l−ε/ 2 <ank<l+ε/2fornk≥N.It follows that
l−ε<an<l+ε forn≥N.Thus the sequence{an}converges with limitl.
Proposition 25 was known to Bolzano (1817) and was clearly stated in the influ-
entialCours d’analyseof Cauchy (1821). However, a rigorous proof was impossible
until the real numbers themselves had been precisely defined.
The M ́eray–Cantor method of constructing the real numbers from the
rationals is based on Proposition 25. We define two fundamental sequences{an}and
{an′}of rational numbers to be equivalent ifan−an′→0asn→∞.Thisisindeed
an equivalence relation, and we define a real number to be an equivalence class of
fundamental sequences. The set of all real numbers acquires the structure of a field if
addition and multiplication are defined by
{an}+{bn}={an+bn}, {an}·{bn}={anbn}.It acquires the structure of a complete ordered field if the fundamental sequence{an}is
said to be positive when it has a positive lower bound. The fieldQof rational numbers
may be regarded as a subfield of the field thus constructed by identifying the ratio-
nal numberawith the equivalence class containing the constant sequence{an},where
an=afor everyn.
It is not difficult to show that an ordered field is complete if every bounded
monotonic sequence is convergent, or ifevery bounded sequence has a convergent
subsequence. In this sense, Propositions 22 and 24 state equivalent forms for the least
upper bound property. This is not true, however, for Proposition 25. An ordered field
need not have the least upper bound property, even though every fundamental sequence
is convergent. It is true, however, that an ordered field has the least upper bound
property if and only if it has the Archimedean property (Proposition 19)andevery
fundamental sequence is convergent.
In a course of real analysis one would now define continuity and prove those
properties of continuous functions which, in the 18th century, were assumed as
‘geometrically obvious’. For example, for givena,b∈Rwitha<b,letI=[a,b]be
theintervalconsisting of allx∈Rsuch thata≤x≤b.Iff:I→Ris continuous,
then it attains its supremum, i.e. there existsc∈Isuch that f(x)≤f(c)for every
x ∈I. Also, iff(a)f(b)<0, thenf(d)=0forsomed ∈I(the intermediate-
value theorem). Real analysis is not our primary concern, however, and we do not feel
obliged to establish even those properties which we may later use.