Number Theory: An Introduction to Mathematics

30 I The Expanding Universe of Numbers

Here the triangle inequality holds in the stronger form

d(a,b)≤max[d(a,c),d(c,b)].

This metric space plays a basic role in the theory ofdynamical systems.

(vii) A connectedgraphcan be given the structure of a metric space by defining the dis-
tance between two vertices to be the number of edges on the shortest path joining them.

LetEbe an arbitrary metric space and{an}a sequence of elements ofE.The
sequence{an}is said toconverge, withlimit a∈E,if

d(an,a)→0asn→∞,

i.e. if for each realε>0 there is a corresponding positive integerN=N(ε)such that
d(an,a)<εfor everyn≥N.
The limitais uniquely determined, since if also d(an,a′)→0, then

d(a,a′)≤d(an,a)+d(an,a′),

and the right side can be made arbitrarily small by takingnsufficiently large. We write

lim

n→∞

an=a,

oran→aasn→∞. If the sequence{an}has limita, then so also does any (infinite)
subsequence.
Ifan→aandbn→b,thend(an,bn)→d(a,b), as one sees by takinga′=an
andb′=bnin(∗).
The sequence{an}is said to be afundamental sequence, or ‘Cauchy sequence’,
if for each realε>0 there is a corresponding positive integerN=N(ε)such that
d(am,an)<εfor allm,n≥N.
If{an}and{bn}are fundamental sequences then, by(∗), the sequence{d(an,bn)}
of real numbers is a fundamental sequence, and therefore convergent.
AsetS⊆ Eis said to beboundedif the set of all real numbers d(a,b)with
a,b∈Sis a bounded subset ofR.
Any fundamental sequence{an}is bounded, since if

d(am,an)<1forallm,n≥N,

then

d(am,an)< 1 +δ for allm,n∈N,

whereδ=max 1 ≤j<k≤Nd(aj,ak).
Furthermore, any convergent sequence{an}is a fundamental sequence, as one sees
by takinga=limn→∞anin the inequality

d(am,an)≤d(am,a)+d(an,a).

A metric space is said to becompleteif, conversely, every fundamental sequence
is convergent.