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Converging sequences

  1. Jul 27, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove or give a counterexample: If {an} and {an + bn} are convergent sequences, then {bn} is a convergent sequence.

    2. The attempt at a solution

    Ok I couldn't think of any counterexamples, so I tried to prove it using delta epsilon definitions:

    |an - L| < E
    |an + bn - M| < E
    want to show: |bn - N| < E

    Is this the right approach?
  2. jcsd
  3. Jul 27, 2008 #2


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    Yes, and M=L+N, right? You can certainly prove that. It's the correct approach.
  4. Jul 27, 2008 #3
    yeah I got the N = M-L part. But then after that I go in circles trying to show it is < Epsilon. =[

    What's the little trick?
  5. Jul 28, 2008 #4
    Hint: [tex] \lim_{n \to \infty} \left\{ a_{n} + b_{n} \right\} = \lim_{n \to \infty} a_{n} + \lim_{n \to \infty} b_{n}[/tex]
  6. Jul 28, 2008 #5


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    And bn=(an+bn)+(-an).
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