Economic growth and economic development 282

Introduction to Modern Economic Growth
Proof. (Existence) Note T n z = T (T n−1 z) for any n = 1, 2, .... Choose z0 ∈ S,

and construct a sequence {zn }∞
n=0 with each element in S, such that zn+1 = T zn so
that

zn = T n z0 .
Since T is a contraction, we have that
d(z2 , z1 ) = d(T z1 , T z0 ) ≤ βd(z1 , z0 ).
Repeating this argument
(6.5)

d(zn+1 , zn ) ≤ β n d(z1 , z0 ),

n = 1, 2, ...

Hence, for any m > n,
d(zm , zn ) ≤ d(zm , zm−1 ) + ... + d(zn+2 , zn+1 ) + d(zn+1 , zn )
Ê
Ô
m1 + ... + n+1 + n d(z1 , z0 )
Ô
Ê
= n mn1 + ... + β + 1 d(z1 , z0 )
βn
d(z1 , z0 ),
≤
1−β
where the first inequality uses the triangle inequality (which is true for any metric
(6.6)

d, recall the Mathematical Appendix). The second inequality uses (6.5). The last
inequality uses the fact that 1/ (1 − β) = 1 + β + β 2 + ... > β m−n−1 + ... + β + 1.

The string of inequalities in (6.6) imply that as n → ∞, m → ∞, zm and zn

will be approaching each other, so that {zn }∞
n=0 is a Cauchy sequence. Since S is

complete, every Cauchy sequence in S has an limit point in S, therefore:
zn → zˆ ∈ S.

The next step is to show that zˆ is a fixed point. Note that for any z0 ∈ S and

any n ∈ N, we have

d(T zˆ, zˆ) ≤ d(T zˆ, T n z0 ) + d(T n z0 , zˆ)

≤ βd(ˆ
z , T n−1 z0 ) + d(T n z0 , zˆ),

where the first relationship again uses the triangle inequality, and the second inequality utilizes the fact that T is a contraction. Since zn → zˆ, both of the terms
on the right tend to zero as n → ∞, which implies that d(T zˆ, zˆ) = 0, and therefore
T zˆ = zˆ, establishing that zˆ is a fixed point.
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