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Sequence and subsequence - real analysis

  1. Feb 22, 2015 #1
    Hello,
    Solving last exam and stuck in this exercise
    1. The problem statement, all variables and given/known data
    Consider an increasing sequence {xn} . We suppose ∃ x∈ℝ and {xnk} a sebsequence of {xn} and xnk→x
    a/ Show that for any n∈ℕ , ∃ k∈ℕ as n≤nk
    b/ Show that xn→x
    2. Relevant equations
    3. The attempt at a solution

    For b/ it is easy.
    But for a/ I really don't know how to do that

    thanks
     
  2. jcsd
  3. Feb 23, 2015 #2

    Stephen Tashi

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    I don't know how to interpret that statement. What is it that happens "as" [itex] n \le n_k [/itex] ?
     
  4. Feb 23, 2015 #3

    HallsofIvy

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    If you mean "n∈ℕ , ∃ k∈ℕ such that n≤nk" that just says that, given any integer n, there exist an "nk", an index from the subsequence, larger than n. And that comes from the fact that the subsequence is infinite.
     
  5. Feb 23, 2015 #4
    Yes sorry it is such that , it was late !
    I dont know where to start from to get to this result ? :oldconfused:
     
  6. Feb 25, 2015 #5

    SammyS

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    Well, what do you mean by nk ?

    Isn't {nk} an increasing sequence in , so that {xnk} is a subsequence ?
     
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