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Convergent sequences

  1. Jun 24, 2011 #1
    If [itex]\{a_n\}\to A, \ \{a_nb_n\}[/itex] converge, and [itex]A\neq 0[/itex], then prove [itex]\{b_n\}[/itex] converges.

    Let [itex]\epsilon>0[/itex]. Then [itex]\exists N_1,N_2\in\mathbb{N}, \ n\geq N_1,N_2[/itex]

    [tex]|a_n-A|<\frac{\epsilon}{2}[/tex]

    And let [itex]\{a_nb_n\}\to AB[/itex]

    So, [itex]|a_nb_n-AB|<\epsilon[/itex]

    I don't know how to show b_n is < epsilon.
     
  2. jcsd
  3. Jun 25, 2011 #2

    tiny-tim

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    Hi Dustinsfl! :smile:

    Hint: an(bn - B) :wink:
     
  4. Jun 25, 2011 #3
    I am don't understand, so we have:

    [tex](a_nb_n-a_nB)[/tex]

    Ok, now what?
     
  5. Jun 26, 2011 #4

    tiny-tim

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    limn->∞ :wink:
     
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