Number Theory: An Introduction to Mathematics

4 Metric Spaces 33

Consequently, ifn>m≥0,

d(xn,xm)≤d(xn,xn− 1 )+d(xn− 1 ,xn− 2 )+···+d(xm+ 1 ,xm)

≤(θn−^1 +θn−^2 +···+θm)d(x 1 ,x 0 )

≤θm( 1 −θ)−^1 d(x 1 ,x 0 ),

since 0<θ<1. It follows that{xn}is a fundamental sequence and so a convergent
sequence, sinceEis complete. Ifx ̄=limn→∞xn,then

d(f(x ̄),x ̄)≤d(f(x ̄),xn+ 1 )+d(xn+ 1 ,x ̄)

≤θd(x ̄,xn)+d(x ̄,xn+ 1 ).

Since the right side can be made less than any given positive real number by takingn
large enough, we must havef(x ̄)= ̄x. The proof shows also that, for anym≥0,

d(x ̄,xm)≤θm( 1 −θ)−^1 d(x 1 ,x 0 ). 

The contraction principle is surprisingly powerful, considering the simplicity of its
proof. We give two significant applications: an inverse function theorem and an exis-
tence theorem for ordinary differential equations. In both cases we will use the notion
of differentiability for functions of several real variables. The unambitious reader may
simply taken=1 in the following discussion (so that ‘invertible’ means ‘nonzero’).
Functions of several variables are important, however, and it is remarkable that the
proper definition of differentiability in this case was first given by Stolz (1887).
Amapφ:U →Rm,whereU ⊆Rnis aneighbourhoodofx 0 ∈Rn(i.e.,U
contains some open ball{x∈Rn:|x−x 0 |<ρ}), is said to bedifferentiableatx 0 if
there exists a linear mapA:Rn→Rmsuch that

|φ(x)−φ(x 0 )−A(x−x 0 )|/|x−x 0 |→0as|x−x 0 |→ 0.

(The inequalities between the various norms show that it is immaterial which norm is
used.) The linear mapA, which is then uniquely determined, is called thederivativeof
φatx 0 and will be denoted byφ′(x 0 ).
This definition is a natural generalization of the usual definition whenm=n=1,
since it says that the differenceφ(x 0 +h)−φ(x 0 )admits the linear approximationAh
for|h|→0.
Evidently, ifφ 1 andφ 2 are differentiable atx 0 ,thensoalsoisφ=φ 1 +φ 2 and

φ′(x 0 )=φ 1 ′(x 0 )+φ 2 ′(x 0 ).

It also follows directly from the definition that derivatives satisfy thechain rule:If
φ:U→Rm,whereUis a neighbourhood ofx 0 ∈Rn, is differentiable atx 0 ,andif
ψ:V→Rl,whereVis a neighbourhood ofy 0 =φ(x 0 )∈Rm, is differentiable aty 0 ,
then the composite mapχ=ψ◦φ:U→Rlis differentiable atx 0 and

χ′(x 0 )=ψ′(y 0 )φ′(x 0 ),

the right side being the composite linear map.