24 I The Expanding Universe of Numbers
The notion of convergence can be defined in any totally ordered set. A sequence
{an}is said toconverge, withlimit l,ifforanyl′,l′′such thatl′<l<l′′, there exists
a positive integerN=N(l′,l′′)such that
l′<an<l′′ for everyn≥N.The limitlof the convergent sequence{an}is clearly uniquely determined; we write
lim
n→∞
an=l,oran→lasn→∞.
It is easily seen that any convergent sequence isbounded, i.e. it has an upper bound
and a lower bound. A trivial example of a convergent sequence is theconstantsequence
{an},wherean=afor everyn; its limit is againa.
In the setRof real numbers, or in any totally ordered set in which each bounded
sequence has a least upper bound and a greatest lower bound, the definition of conver-
gence can be reformulated. For, let{an}be a bounded sequence. Then, for any positive
integerm, the subsequence{an}n≥mhas a greatest lower boundbmand a least upper
boundcm:
bm=inf
n≥m
an, cm=sup
n≥man.The sequences{bm}m≥ 1 and{cm}m≥ 1 are also bounded and, for any positive integerm,
bm≤bm+ 1 ≤cm+ 1 ≤cm.If we define thelower limitandupper limitof the sequence{an}by
lim
n→∞an:=sup
m≥ 1bm, lim
n→∞
an:=inf
m≥ 1cm,then limn→∞an≤limn→∞an, and it is readily shown that limn→∞an=lif and only
if
lim
n→∞an=l= lim
n→∞
an.A sequence{an}is said to benondecreasingifan≤an+ 1 for everynandnonin-
creasingifan+ 1 ≤anfor everyn.Itissaidtobemonotonicif it is either nondecreasing
or nonincreasing.
Proposition 22Any bounded monotonic sequence of real numbers is convergent.
Proof Let{an}be a bounded monotonic sequence and suppose, for definiteness, that
it is nondecreasing:a 1 ≤a 2 ≤a 3 ≤···. In this case, in the notation used above we
havebm=amandcm=c 1 for everym. Hence
lim
n→∞an=sup
m≥ 1am=c 1 = lim
n→∞
an. Proposition 22 may be applied to the centuries-old algorithm for calculating square
roots, which is commonly used today in pocket calculators. Take any real number
a>1 and put
x 1 =( 1 +a)/ 2.