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Proving Convergence to 0

  1. Sep 29, 2012 #1
    1. The problem statement, all variables and given/known data

    If sequence an diverges to infinity and sequence an*bn converges then how do I prove that sequence bn must converge to zero?


    2. Relevant equations



    3. The attempt at a solution
    I really don't know how to go about this so any help would be so appreciated.
    Thanks
     
  2. jcsd
  3. Sep 29, 2012 #2

    HallsofIvy

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    If [itex]a_nb_n[/itex] converges, to, say L, then, given any [itex]\epsilon> 0[/itex], there exist [itex]N_1[/itex] such that if [itex]n> N_1[/itex], [itex]|a_nb_n- L|< \epsilon[/itex].

    Since [itex]a_n[/itex] diverges to infinity, then, given any X>0, there exist [itex]N_2[/itex] such that if [itex]n> N_2[/itex], [itex]a_n> X[/itex].

    Take n greater than the larger of [itex]N_1[/itex] and [itex]N_2[/itex] and use both of those.
     
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