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How can i prove these two convergence theorem?

  1. Oct 26, 2011 #1
    1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent

    2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ...., then {an/bn} is also convergen
     
  2. jcsd
  3. Oct 26, 2011 #2

    CompuChip

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    Well, given that an converges to a and bn converges to b, you could "guess" the limits of the other series and then try to prove that they actually are the limits from the definition.

    E.g. you can try to show that
    [tex]\forall_{\epsilon > 0} \exists_{N_+} : n > N_+ \implies |(a_n + b_n) - L| < \epsilon[/tex]
    where L = a + b is the postulated limit. Of course you already know that given such an [itex]\epsilon[/itex] you can find Na and Nb such that
    [tex]n > N_a \implies |a_n - a| < \epsilon[/tex]
    and
    [tex]n > N_b \implies |b_n - b| < \epsilon[/tex]
     
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