Question:

Let g : be a function such that the following holds:

There is a constant 0 ≤ L ≤ 1 such that

| g (x) – g (y) | ≤ L |x – y| for all x, y .

The aim of this problem is to prove that there is a unique x* with the property x* = g (x*) (such an x* is called a fixed point of g). To do this, we will follow the following scheme:

First choose an arbitrary, but fixed, x₀ . Then define inductively

x_n+1 = g (x_n) for every integer n ≥ 0.

a) Prove that for every n ≥ 1, we have

|x_n+1 - x_n| ≤ Lⁿ |x₁ - x₀ |

b) Conclude from the inequality (0.1) that the sequence {x_n} is a Cauchy sequence

c) Define now

x* =

Note that the limit exists because {x_n} is Cauchy. Prove that x* = g (x*)

(Hint: Recall the definition of the sequence x_n)

d) Prove that x* is the only real number with the property t = g(t)

(Hint: Assume that there are x₁, x₂ with x_{1 =} g (x₁) and x₂ = g (x₂) and use the fact that 0 ≤ L ≤ 1 to show that x₁ = x₂₎