1. ## Contraction Mapping Principle

I am given this as one of the problems that might be on the exam. We need to know how to prove it, and I don't remember it from class. We haven't gone over metric spaces, whatever they are, so I don't know why it's given as a problem. Anyway, can someone outline the proof for me?

2. Originally Posted by anon2194
I am given this as one of the problems that might be on the exam. We need to know how to prove it, and I don't remember it from class. We haven't gone over metric spaces, whatever they are, so I don't know why it's given as a problem. Anyway, can someone outline the proof for me?

You could google "Brouwer's fixed point theorem" or something like that...anyway, we're in a complete metric space, take any point $\displaystyle x_0$ and then define $\displaystyle x_n:= f^{n-1}(x_0)$ , with $\displaystyle f^{1}(x_0)=f, f^{2}(x_0)=f(f(x_0)),...,f^{n}(x_0)=f^{n-1}(f(x_0))$.
Now show $\displaystyle \{x_n\}$ is a Cauchy sequence and thus it converges to some point $\displaystyle x_1$, and this point is a fixed point of the function (of course, you need here continuity of f...)

Tonio

Tonio

3. I remember now. We went over the one-dimensional case. I just need to use the Intermediate Value Theorem for that one. Thanks

4. Do u need proof? If tonio don't mind..

5. No, I have the proof. Wikipedia told me to use g(x) = f(x) - x for f:[a,b]->[a,b] and the Intermediate Value Theorem to find a g(x) = 0, etc etc