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Proving Convergence to 0

  • Thread starter sitia
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  • #1
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Homework Statement



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


Homework Equations





The Attempt at a Solution


I really don't know how to go about this so any help would be so appreciated.
Thanks
 

Answers and Replies

  • #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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