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Math Help - Pointwise vs Uniform Convergence

  1. #1
    Ant
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    Pointwise vs Uniform Convergence

    I've been trying, with no luck so far, to get my head around the differnce between pointwise and uniform convergence.


    The definition which I am using for Uniform convergence is:

    We say  f_{m} converges to f uniformly on a set A if

     | f(z) - f_{m}(z)| \leq C_{m},\forall z \in A\\

    Where  C_{m} are some constants such that C_{m} \to 0 as m \to \infty

    I understand this definition, but what I am having trouble with is seeing how any convergence sequence of function could fail to satisfy this!


    Assume  f_{m}(z) \to f(z) but not uniformly.

    Now we fix m and calculate | f(z) - f_{m}(z)| for each possible choice of  z \in A. There must be a value for z which gives the greatest value (i.e. for which f_{m}(z)is furthest away from f(z). So take this value and denote is C_m for each choice m. Then for each m we have certainly satisfied | f(z) - f_{m}(z)| \leq C_{m}, \forall z \in A\\ .

    Moreover this sequence   C_{m} must converge to 0 as have by assumption that  f_{m}(z) \to f(z) . So  f_{m} converges to f uniformly on a set A. Contradicting our assumption that it converged but not uniformly!

    Clearly there must be something very wrong with the above argument but I can't see what it is.

    thanks for any help or explanation!
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  2. #2
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    Re: Pointwise vs Uniform Convergence

    Here is a counter example for you to think about.

    f_n(x)=\begin{cases}1, \text{ if } |x| \le n \\ 0, \text{ if } |x| > n\end{cases}

    Pointwise f_n(x) \to 1

    Note that |f(x)-f_n(x)|=1 for all n.
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  3. #3
    Ant
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    Re: Pointwise vs Uniform Convergence

    Is  f(x) = 1 above?

    Also, is the domain just:  x \in Real numbers ?

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