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Help on Sequence

  1. Feb 20, 2007 #1
    1. The problem statement, all variables and given/known data
    The question asks to prove that if (t_n) is a convergent sequence and suppose that its limit is great than a number x. The prove that it exists a number N such that n>N => t_n>x

    3. The attempt at a solution
    I tried to say that as (t_n) converge, (t_n - x) converges too.
    And let lim(t_n - x) = k, for k in R (real).
    Then for some e>0, there's a number N s.t. n>N => |(t_n-x)-k|< e
    Take e=0, then |(t_n-x)-k|=0 => t_n-k = x,
    since lim(t_n)>x then k must be positive => lim(t_n)>x.

    can anyone tell me if i am going the wrong way??
    I am afraid that my concept somewhere is wrong..
    Thanks
     
  2. jcsd
  3. Feb 20, 2007 #2

    NateTG

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    Generally speaking, when you write epsilon proofs, it's easier to follow the argument of the proof if you start by specifying what you want [itex]\epsilon[/itex] to be. (Even you worked your way back to it.)

    You can't use [itex]\epsilon = 0[/itex] since that contradicts [itex]\epsilon > 0[/itex].

    You need to be much more careful in the way you write things in general.
     
  4. Feb 20, 2007 #3
    Then can i rephrase it as follows:
    As (t_n) converge, (t_n - x) converges too.
    And let lim(t_n - x) = k, for k in R, k>0.
    Then for some e>0, there's a number N s.t. n>N => 0<|(t_n-x)-k|< e
    |(t_n-x)-k|>0 implies t_n - k>x, since k is positive, it follows that t_n>x.

    does this sound right? (it seems like i am ignoring the use of e?)
    if not, can you give me some idea on how to make the proof works?
    Thanks.
     
  5. Feb 20, 2007 #4

    HallsofIvy

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    If L is the limit and L> x, then L-x> 0. What if you make [itex]\epsilon= (L-x)/2[/itex]?
     
  6. Feb 21, 2007 #5
    Thanks for replying!
    When I have 2 situations, how can i show that when L-tn>0 (L>tn) and L>x actually implies tn>x?? Can you give me some idea?
     
  7. Feb 21, 2007 #6

    HallsofIvy

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    All I can do is repeat: if the limit L> x, then L-x> 0. What happens if you take [itex]\epsilon= (L-x)/2[/itex]?
     
  8. Feb 21, 2007 #7

    NateTG

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    Well, it might also be hand to review the definition of convergent sequence....
     
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