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Thread: Show it converges uniformly

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

    Suppose that g : R $\displaystyle \rightarrow$ R is such that there exists r $\displaystyle \in$ (0,1) with the property that |g'(x)| $\displaystyle \leq$ r for every x $\displaystyle \in$ R. Let $\displaystyle F_{0}$ : R $\displaystyle \rightarrow$ R be a bounded function and define recursively $\displaystyle F_{n}(x)$ := $\displaystyle g(F_{n-1}(x))$, for each n $\displaystyle \geq$ 1. Show that {$\displaystyle F_{n}$}$\displaystyle _{n \in N}$ 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 $\displaystyle \rightarrow$ R is such that there exists r $\displaystyle \in$ (0,1) with the property that |g'(x)| $\displaystyle \leq$ r for every x $\displaystyle \in$ R. Let $\displaystyle F_{0}$ : R $\displaystyle \rightarrow$ R be a bounded function and define recursively $\displaystyle F_{n}(x)$ := $\displaystyle g(F_{n-1}(x))$, for each n $\displaystyle \geq$ 1. Show that {$\displaystyle F_{n}$}$\displaystyle _{n \in N}$ converges uniformly on R.
    This is similar to the contraction mapping principle.

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

    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.

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

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

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

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

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

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

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

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