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Functional analysis and limits

  1. Oct 6, 2008 #1
    1. The problem statement, all variables and given/known data

    [​IMG]

    2. Relevant equations
    [tex] \lim_n a_n := \lim_{n \rightarrow \infty} a_n [/tex]


    3. The attempt at a solution
    I'm stuck at exercise (c). Since if n heads to infinity the m doesn't play the role the limit must be one. So the mistake is somewhere on the left and I think it is at the part where both limits are taken at the same time.

    Or is the limit 1/2? Can someone help me?
     
  2. jcsd
  3. Oct 6, 2008 #2

    Dick

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    Does f_n really converge to 1? What's ||f_n-1|| for any n?
     
  4. Oct 6, 2008 #3

    morphism

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    Hint: we have to be careful when we say things like "[itex]f_n \to \textbf{1}[/tex]" nonchalantly. What does "[itex]\to[/itex]" mean in this setting?
     
  5. Oct 7, 2008 #4

    [tex] ||f_n-1|| = \frac{m}{m+n} [/tex] for any n. But if I let n tend to infinity this would go to zero, right? So the limit is one? I guess I miss the point here because the n stands for the nth sequence and the m is just the element in that sequence, right?

    If I look in one sequence (so for fixed n) and let m tend to infinity I would always get zero do you mean that?

    It means keeping m fixed while letting [tex] n \rightarrow \infty [/tex] right?
     
    Last edited: Oct 7, 2008
  6. Oct 7, 2008 #5

    Dick

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    Don't take n to infinity. Just focus on a single n for a minute. I think the limit m->infinity of m/(m+n) is one, not zero. So I would say ||f_n-1||=1. For ALL n. So I would say while it converges pointwise, the sequence f_n does not converge in l_infinity.
     
  7. Oct 7, 2008 #6
    So there holds: [tex] \lim_{m \rightarrow \infty} f_n =1 [/tex] imlpying pointwise convergence.

    A sequence converges in [tex]l^{\infty}[/tex] if [tex] \lim_{j \rightarrow} ||f_j-f||_{\infty} \rightarrow 0[/tex] but by noticing the pointwise convergence one can note that this will not head to zero but 1, right?
     
    Last edited: Oct 7, 2008
  8. Oct 7, 2008 #7

    Dick

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    I think you've got the right idea. I think your first statement should say lim n->infinity f_nm=1. Each term m of the sequences tends to 1 as n->infinity. But that's not enough to make the f_n converge.
     
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