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Sequence question.

  1. Nov 15, 2006 #1

    MathematicalPhysicist

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    i have that lim x_n=lim y_n=a
    and we have the sequence (x1,y1,x2,y2,....)
    i need to show that this sequence (let's call it a_n) converges to a.

    well in order to prove it i know that if a_n has a unique partial limit then it converges to it to it, but how do i show here that it has a unique partial limit?
    i mean if we take the subsequences in the even places or the odd places then obviously those subsequences converges to the same a, but i need to show this is true for every subsequence, how to do it?
    thanks in advance.
     
  2. jcsd
  3. Nov 15, 2006 #2

    HallsofIvy

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    I think you are making too much of this. Since xn converges to a, given [itex]\epsilon> 0[/itex] there exist N1 such that if n> N1 then [itex]|x_n-a|< \epsilon[/itex]. Since yn converges to a, given [itex]\epsilon> 0[/itex] there exist N2 such that if n> N1 then [itex]|y_n-a|< \epsilon[/itex].

    What happens if you take N= max(N1, N2)?
     
  4. Nov 15, 2006 #3

    MathematicalPhysicist

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    yes i see your point.
    if we take the max of the indexes then from there, we have that either way a_n equals x_n or y_n, and thus also a_n converges to a, cause from the maximum of the indexes both the inequalities are applied and thus also |a_n-a|<e, right?
     
  5. Nov 15, 2006 #4

    HallsofIvy

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    Yes, that is correct.
     
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