Exercise 9
.
17(6) (Contraction mapping)
Patrick De Leenheer
February 18, 2005
Proposition 1.
Let
f
:
I
→
I
be continuous on the closed interval
I
and assume that there exists
0
< α <
1
such that:
∀
x, y
∈
I
:

f
(
x
)

f
(
y
)
 ≤
α

x

y

(1)
Then
f
is continuous.
Pick
x
1
∈
I
and construct the sequence
{
x
n
}
as follows:
x
n
=
f
(
x
n

1
)
,
∀
n >
1
.
Then there is some
x
*
∈
I
such that
x
n
→
x
*
as
n
→ ∞
. Moreover,
f
(
x
*
) =
x
*
(that is,
f
has a
Fxed point in
I
).
Proof.
Fix
x, y
∈
I
and
² >
0, and choose
δ
=
²/α
.
Then if

x

y

< δ
, it follows from (1) that

f
(
x
)

f
(
y
)
 ≤
α

x

y

and thus that

f
(
x
)

f
(
y
)

< αδ
=
²
. This implies that
f
is continuous
on
I
.
Pick
x
1
∈
I
and construct the sequence
{
x
n
}
as outlined above.
We will show that
{
x
n
}
is a
Cauchy sequence.
If we do that, then it follows that
x
n
→
x
*
as
n
→ ∞
for some
x
*
.
Since
I
is
closed, there must hold that
x
*
∈
I
. Now, since
f
is continuous on
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 Summer '06
 DeLeenheer
 Metric space, Xn, Patrick De Leenheer

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