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, x ₀ . Then define inductively

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

1. Prove that for every n ≥ 1, we have
   |x _n+1 - x _n | ≤ L ⁿ |x ₁ - x ₀ |
2. Conclude from the inequality (0.1) that the sequence {x _n } is a Cauchy sequence
3. 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 )
4. Prove that x* is the only real number with the property t = g(t)

(Hint: Assume that there are x ₁ , x ₂ with x _{1 =} g (x ₁ ) and x ₂ = g (x ₂ ) and use the fact that 0 ≤ L ≤ 1 to show that x ₁ = x ₂₎

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