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Math Help - Show it converges uniformly

  1. #1
    Member thaopanda's Avatar
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    Show it converges uniformly

    Suppose that g : R \rightarrow R is such that there exists r \in (0,1) with the property that |g'(x)| \leq r for every x \in R. Let F_{0} : R \rightarrow R be a bounded function and define recursively F_{n}(x) := g(F_{n-1}(x)), for each n \geq 1. Show that { F_{n}} _{n \in N} converges uniformly on R.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    Suppose that g : R \rightarrow R is such that there exists r \in (0,1) with the property that |g'(x)| \leq r for every x \in R. Let F_{0} : R \rightarrow R be a bounded function and define recursively F_{n}(x) := g(F_{n-1}(x)), for each n \geq 1. Show that { F_{n}} _{n \in N} converges uniformly on R.
    This is similar to the contraction mapping principle.

    \frac{|g(x)-g(y)|}{|x-y|}=|g'(c)|\leq r\implies |g(x)-g(y)|\leq r|x-y|

    Have you learned the CMP yet?
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  3. #3
    Member thaopanda's Avatar
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    I don't think I've learned that yet...
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    This is similar to the contraction mapping principle.

    \frac{|g(x)-g(y)|}{|x-y|}=|g'(c)|\leq r\implies |g(x)-g(y)|\leq r|x-y|

    Have you learned the CMP yet?
    |F_{n+1}-F_n|=|g(F_n)-g(F_{n-1})|\leq r|F_n-F_{n-1}|= r|g(F_{n-1})-g(F_{n-2})|\leq r^2|F_{n-1}-F_{n-2}|\leq...\leq r^n|F_1-F_0| (You might want to prove this using induction.)

    Let d=|F_1-F_0|. For m>n, by the triangle inequality,

    |F_m-F_n|\leq |F_m-F_{m-1}|+|F_{m-1}+F_{m-2}|+...+|F_{n+1}-F_n|\leq r^{m-1}d+r^{m-2}d+...+r^nd= dr^n\sum_{k=0}^{m-n-1}r^k\leq dr^n\sum_{k=0}^{\infty}r^k=\frac{dr^n}{1-r} (You might also want to prove this by induction.)

    Since r<1 (and d is bounded), \forall \epsilon>0, \exists N such that n>N implies r^n<\frac{\epsilon(1-r)}{d}. So for all \epsilon>0, m>n>N implies

    |F_m-F_n|\leq\frac{dr^n}{1-r}<\frac{d}{1-r}\cdot\frac{\epsilon(1-r)}{d}=\epsilon

    so \{F_n\} converges uniformly by the Cauchy criterion.

    Note: The above proof I gave is basically the proof of the CMP.
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