Subsequential limit question

  1. 1. The problem statement, all variables and given/known data
    Suppose {a_n} is a bounded sequence who's set of all subsequential
    limits points is {0,1}. Prove that there exists two subsequences,
    such that: one subsequence converges to 1 while the other converges
    to 0, and each a_n belongs to exactly one of these subsequences.


    2. Relevant equations



    3. The attempt at a solution
    Well, it's clear that at the limit points 0 and 1; there is a subsequence that that converges to it. I'm not quite sure about how to prove that each a_n belongs to exactly one of these subsequences or how to apply the bounded property of {a_n} into this question.
     
  2. jcsd
  3. lurflurf

    lurflurf 2,327
    Homework Helper

    This will depend slightly on the definition you are using.
    Suppose {a_n} is a bounded sequence who's set of all subsequential
    limits points is {0,1}
    Suppose epps is a positive real number
    0 and one are limit points so it is known that
    (-eps,eps)U(1-eps,1+eps)
    Contains all but a finite number of the a_n
    now the sequence can be easily partitioned by
    1/2
    one subsequence if a_n<=1/2
    another if a_n>1/2
     
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