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Math Help - prove that :

  1. #1
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    prove that :

      {Let \ x _n \ be \ a \ sequence \ of \ integers \ ; \ x_n \in Z \ , \ \forall n \in N   }   { . If \ x_n \ is \ cauchy \ , \ prove \ that \ x_n \ is  \ eventually \ constant \ }
     {(i.e , \exists c\in Z , \exists k \in N  \ s.t \ x_n =c , \forall n \geq k )}
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  2. #2
    Super Member girdav's Avatar
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    Take \epsilon =\frac 12 in the definition of a Cauchy sequence.
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  3. #3
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    Assume that the  x_n don't eventually become constant. That is, for all integers k,  |x_i - x_j| > 0 for some integer i and some integer j greater than k.

    Since the sequence is Cauchy, there is an integer k such that  |x_n - x_m| < \frac{1}{2} for all n and m greater than k. From our assumption, we can take n = i and m = j. Then  0 < |x_i - x_j| < \frac{1}{2} . This obviously can't be true, as the difference between two integers must be an integer itself. QED
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