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Math Help - Contraction Mapping Principle

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by anon2194 View Post
    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 x_0 and then define x_n:= f^{n-1}(x_0) , with 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 \{x_n\} is a Cauchy sequence and thus it converges to some point x_1, and this point is a fixed point of the function (of course, you need here continuity of f...)

    Tonio

    Tonio
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  3. #3
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    I remember now. We went over the one-dimensional case. I just need to use the Intermediate Value Theorem for that one. Thanks
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  4. #4
    Member kjchauhan's Avatar
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    Do u need proof? If tonio don't mind..
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  5. #5
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    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
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