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Math Help - uniform convergence

  1. #1
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    uniform convergence

    i have a question about uniform convergence. definition of uniform convergence is given \epsilon >0, there exists N such that for all x and for all n \geq N, we have |f_n (x) -f(x)|<\epsilon. i was wondering why the pointwise convergence does not imply uniform convergence. i know that for pointwise convergence, depending on x, you have to find a different N to have |f(x)-f_n(x)| < \epsilon. but for each x, f_n will eventually be within the distance of \epsilon for n \geq N_x for some natural number N_x so if you choose max of such N_x and call it M, then can we say for given \epsilon >0, for all x and for all n \geq M |f_n(x)-f(x)|< \epsilon?
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  2. #2
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    It might be that \sup N_x=\infty. Just consider f_n(x)=1 if x\in [n,n+1] and 0 otherwise. Then f_n\to 0, but it isn't uniform.
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  3. #3
    Senior Member Shanks's Avatar
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    max of N_x?
    it is possible that N_x is not bounded over all x.
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