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Homework Help: Convergent Sequences on l infinity

  1. Jun 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Define [itex] R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \} [/itex]

    Define [itex] l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} | t^{(k})| < \infty \} [/itex]

    Observe that [itex] R^\infty_f [/itex] is a linear subspace of [itex] l^\infty[/itex]. Show that [itex] R^\infty_f [/itex] is not closed in [itex] l^\infty[/itex], then show that the closure of [itex] R^\infty_f [/itex] is the space [itex] c_0 [/itex];

    2. Relevant equations

    The space c_0 is the set of all sequences converging to zero


    3. The attempt at a solution

    It's not too hard to show that this set is not closed. It suffices to show that there is a convergent sequence in [itex] l^\infty[/itex] such that every term is in [itex] R^\infty_f [/itex], but whose limit is not in [itex] R^\infty_f [/itex]. I constructed the following sequence

    [itex] x_1 = (1, 0, \ldots, ) [/itex]
    [itex] x_2 = (1, \frac{1}{2}, 0 , \ldots, ) [/itex]
    [itex] \vdots [/itex]
    [itex] x_n = (1, \ldots, \frac{1}{n}, 0, \ldots} ) [/itex]

    which converges to the point [itex] a = (1, \frac{1}{2}, \ldots, \frac{1}{n-1}, \frac{1}{n}, \frac{1}{n+1}, \ldots ) [/itex]

    It's the closure part that I'm worried about. I'm not terribly sure how I would go about showing that...
     
  2. jcsd
  3. Jun 21, 2008 #2

    morphism

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    It will suffice to show that every sequence in c_0 is a limit of sequences in R_f. Do you agree?
     
  4. Jun 21, 2008 #3
    Yes, since every point in c_0 will necessarily be the limit of some sequence in R_f. Though I think that this only shows that c_0 is a subset of the closure - not necessarily the whole closure.
     
  5. Jun 21, 2008 #4

    morphism

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    Yes, but on the other hand, R_f clearly sits in c_0 (and c_0 is closed!).
     
  6. Jun 21, 2008 #5
    True enough.

    So I need to show that every sequence in c_0 is a limit of sequences in R_f.

    How do I show that every sequence that converges to zero is the limit of a sequence. Indeed, what does it mean for a sequence to be a limit of another sequence?
     
  7. Jun 21, 2008 #6

    morphism

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    Think of this as a problem set in an abstract normed space. What does it mean for a sequence {x_n} to converge to x?
     
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