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Uniform and Pointwise Convergance of Functions

  1. Apr 18, 2009 #1
    Can anyone explain to me why if a sequence of functions {f_n} is uniformly convergent, then it must converge to it's pointwise limit?

    So, if we showed {f_n} is not uniformly convergent to some f (f is the pointwise limit), how do we know that there isn't some other function, say g, such that {f_n} converges to g uniformly?
     
  2. jcsd
  3. Apr 18, 2009 #2

    CompuChip

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    The reason is that uniform convergence is stronger than pointwise convergence.

    Suppose that the limit is f.
    In the definition, you first pick some ε > 0.
    Roughly speaking, for pointwise convergence you can find for any x some number N = N(x) such that for n > N, |fn(x) - f(x)| < ε.
    For uniform convergence you can find an N' which works for all x at the same time. So if you have uniform convergence and you have your ε you can find some N', and if you take N(x) = N' for all x you can prove pointwise convergence.
     
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