- #1

fluidistic

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## Homework Statement

I must understand the proof that if [itex]F:[a,b] \to [a,b][/itex] and [itex]F[/itex] is contractive then there exist a unique [itex]x \in [a,b][/itex] such that [itex]F(x)=x[/itex].

## Homework Equations

Definition of a contractive function: F is contractive over [a,b] if and only if there exist [itex]\lambda[/itex] such that [itex]0<\lambda <1[/itex] and [itex]|F(y)-F(x)| \leq \lambda |y-x| \forall x[/itex] and [itex]y \in [a,b][/itex]. That's from my memory.

## The Attempt at a Solution

I'm almost done understanding the existence (I already have the uniqueness proof and I understand it).

I'm stuck at understanding the latest part.

Here is the -watered down- proof:

Let [itex]x_{n+1}=F(x_n)[/itex] for [itex]n \geq 1[/itex].

[itex]\exists \lambda \in (0,1)[/itex] such that [itex]|x_n-x_{n+1}|=|F(x_{n+1})-F(x_{n+2})| \leq \lambda |x_{n+1}-x_{n+2}| \leq ... \leq \lambda ^{n-1} |x_1-x_0|[/itex].

I can take [itex]x_n=x_0+S_n[/itex] where [itex]S_n=\sum _{j=1}^{\infty } (x_j-x_{j-1})[/itex].

If [itex]\{ S_n \}[/itex] converges, then so do [itex]\{ x_n \}[/itex].

But [itex]\{ S_n \}[/itex] does converge, since it's lesser than [itex]|x_1-x_0|\sum _{j=0} \lambda ^{j-1}[/itex] which is convergent.

Thus [itex]\lim _{n \to \infty} x_n=x[/itex].

Now I must show that [itex]x=F(x)[/itex].

[itex]x=\lim _{n \to \infty} x_n=\lim _{n \to \infty} F(x_{n-1})=F(\lim _{n \to \infty} x_{n-1}) =F(x)[/itex]. Thus [itex]x=F(x)[/itex]. Notice here that the proof used the fact that F is continuous without ever demonstrating it. So it seems that if F is contractive over an interval, it is continuous.

Now the proof wants to show that [itex]x \in [a,b][/itex].

And here comes the part that doesn't make any sense to me.

"Furthermore, since [itex]x\in [a,b][/itex], it follows that [itex]x_n \in [a,b] \forall n[/itex], for [itex]F([a,b])[/itex] is included in [itex][a,b][/itex].

Since [itex][a,b][/itex] is closed in [itex]\mathbb{R}[/itex], F is continuous and [itex]\lim _{n \to \infty} x_n =x[/itex], it follows that [itex]x\in [a,b][/itex]."

My problem is:

The first line assumes what it wants to prove and I don't see the point of the comment that follows.

For the second line, I do not understand the implication. Is there any theorem that justifies the implication? It's not at all intuitive to me.

I'd like an explanation on how to end the proof. I mean, how to show that [itex]x \in [a,b][/itex].

Thanks in advance.