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Real Analysis

  1. Apr 23, 2014 #1
    I have a super round about way to prove this, but I'm having trouble finding a succinct proof

    Let (tn) be diverge and (sn) converge. Show (tn+sn) diverges

    The way I was doing involved considering that tn was unbounded, then showing it (sn+tn) is divergent.

    Then I had to consider that tn is bounded and oscillatory, consider convergent subsequences, and show (sn+tn) had no unique limit, and therefore diverges. This part of the proof seemed less clear and I'm not sure if I can assert that because I have convergent subsequences of (tn) with multiple limits that (sn+tn) also has multiple limits.

    I figure there has got to be some simple contradiction proof involving some triangle inequality trick that I'm just missing.

    Thanks
     
  2. jcsd
  3. Apr 24, 2014 #2

    jbunniii

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    Hint: the sum (or difference) of two convergent sequences is convergent.
     
    Last edited: Apr 24, 2014
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