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Limit of a Sequence

  1. Oct 4, 2011 #1
    The problem statement, all variables and given/known data

    What is the fastest way to prove this.
    1/an→1/a, where an is a sequence.

    The attempt at a solution

    I know how to prove this but I am looking for a simple and elegant proof.
     
  2. jcsd
  3. Oct 4, 2011 #2

    LCKurtz

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    Of course, it isn't true the way you stated it. You need some hypotheses.

    Elegant might mean noting that if a ≠ 0 then 1/x is continuous at a. Whether that is "simple" likely depends on the context.
     
  4. Oct 5, 2011 #3
    an is a sequence. I am trying to prove this limit law for the sequence.
     
  5. Oct 5, 2011 #4

    LCKurtz

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    You haven't even stated the limit law correctly yet. And I thought you said you already know how to prove it. :confused:
     
  6. Oct 5, 2011 #5
    I just abriviated limn→∞ an = a as an→a (as n→∞) if that is what you mean.
     
  7. Oct 5, 2011 #6
    Just to be clear, you're saying that if an goes to a, you want to prove (quickly) that 1/an goes to 1/a? Correct?
     
  8. Oct 5, 2011 #7

    LCKurtz

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    No, I'm not talking about notation. I'm talking about the fact that you haven't stated the theorem correctly even yet. You need something in the form

    If [hypotheses here] then [conclusion here].

    Your original statement, highlighted above, not only doesn't do that, it is false.
     
  9. Oct 5, 2011 #8
    What do you mean it is false? How can a theorem be false? It has been proven. It is part of the Algebraic Limit Theorem.

    1/an→1/a, where an is a sequence and a ≠ 0.
     
  10. Oct 5, 2011 #9

    gb7nash

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    You're stating some kind of conclusion. What's the hypothesis?
     
    Last edited: Oct 5, 2011
  11. Oct 5, 2011 #10

    LCKurtz

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    Did you even read my post?
     
  12. Oct 5, 2011 #11
    Let lim an = a, and lim bn = b. Then, lim an/bn = a/b.
    We know lim (anbn) = ab. So ...
     
  13. Oct 5, 2011 #12
    Never mind I found the neatest way to prove it.
     
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