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A finite set and convergence

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Let A be a finite subset of R. For each n in N, let x_n be in A. Show that if the sequence x_n is convergent then it must become a constant sequence after a while.

    2. Relevant equations

    The definition of limit.

    3. The attempt at a solution

    As A is finite, at least one element of A will appear in the sequence more than once after some N. As this sequence is convergent there is an M for any ε such that |x_n - x| < ε for every n with n>M. Let M>N.... Help please.

    I can't decide whether this problem is too hard or I'm stupid or this is just because I'm a beginner?
     
  2. jcsd
  3. Nov 3, 2012 #2

    HallsofIvy

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    If A= {x_n} is finite then the set of distances {|x_i- x_j|} is finite and so has a smallest value. If [itex]\epsilon[/itex] is smaller than that ...
     
  4. Nov 3, 2012 #3
    Ah okay, thank you
     
  5. Nov 3, 2012 #4
    Idea: Establish a bijection f: N -> A
    n |-> f(n)=x_{n}
    If there exists no N: [itex]\forall n > N, x_{n} = const[/itex], then A must be infinite -> hence we obtain a contradiction. :smile:
     
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