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Real analysis

  1. Nov 2, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.


    2. The attempt at a solution

    Is this correct? Should I add more detail? Thanks for your help!

    Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to f##. We must show that for ##\epsilon>0##, there exists an ##N## such that ##d(f,f^n)<\epsilon## whenever ##n>N## for all ##x.##

    For ##a\in (0,1)##, it is clear to see that ##x^n\to 0## as ##n## approaches infinity. We must then show ##|x^n|<\epsilon## whenever ##n## is greater than some ##N##.

    On ##[0,a]##, ##x^n## attains its max at ##x=a## so ##x^n<a^n##. Then note that ##a^n## decreases with increasing ##n##, so we choose ##N## such that ##a^N<\epsilon##.

    ##\{f_n\}## does not converge uniformly on ##[0,1]## because at ##x=1##, ##f^n = (1)^n =1\ne 0## for all ##n##.
     
    Last edited: Nov 2, 2013
  2. jcsd
  3. Nov 2, 2013 #2

    LCKurtz

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    You need more for the case ##x\in [0,1]##. You haven't shown that the convergence of ##x^n## to the function$$
    f(x) = \left\{\begin{array}{rl}
    0,& 0\le x<1\\
    1, & x=1
    \end{array}\right.$$is not uniform. After all, that is the function to which it converges for each ##x##. Why isn't the convergence uniform?
     
  4. Nov 2, 2013 #3
    LCKurtz - Can you elaborate?

    For ##[0,1]##, ##\{f_n\}## does not converge uniformly on ##[0,1]## because at ##x=1##, ##f^n = (1)^n =1\ne 0## for all ##n##.
     
    Last edited: Nov 2, 2013
  5. Nov 2, 2013 #4

    LCKurtz

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    You know the definition of uniform convergence. Given ##\epsilon > 0## you can find ##N## such that if ##n>N##, ##\|f_n-f\| < \epsilon##. To show this is false you need to find an ##\epsilon## for which you can't find an ##N## that works. You already know the problem is near ##x=1##. Think about that.
     
    Last edited: Nov 2, 2013
  6. Nov 2, 2013 #5
    Okay, thanks.
     
  7. Nov 2, 2013 #6

    LCKurtz

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    Please note I edited my post. It's late and the first one needed fixing.
     
  8. Nov 3, 2013 #7
    Its okay, thanks for the help!
     
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