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Math Help - unbounded sequence

  1. #1
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    unbounded sequence

    let  x_n be an unbounded sequence . prove that there is a subsequence  x_{n_k} of  x_n such that  \frac{1}{x_{n_k}} \to 0
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  2. #2
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    If epsilon > 0 then 1/x_n - 0 < epsilon => x_n > 1/epsilon. If you choose a strictly increasing subsequence of {x_n} (which is possible since it is unbounded), then all but a finite number of x_n > 1/epsilon, so 1/x_n converges to 0.
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  3. #3
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    Quote Originally Posted by flower3 View Post
    let  x_n be an unbounded sequence . prove that there is a subsequence  x_{n_k} of  x_n such that  \frac{1}{x_{n_k}} \to 0
    Suppose that \left(x_n\right) is not bounded above.
    Then  \left( {\exists x_{N_1 } } \right)\left[ {1 < x_{N_1 } } \right].
    Let B_1  = \max \left\{ {2,N_1 } \right\} then  \left( {\exists x_{N_2 } } \right)\left[ {B_1 < x_{N_2 } } \right].
    Now you can see how to construct the required subsequence.
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