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Math Help - Convergence Question

  1. #1
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    Convergence Question

    I need help proving this question:

    Suppose lim(n->oo) An = L and let Bn = A2n for all n being Natural numbers. Prove that lim(n->oo) Bn = L
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Porter1 View Post
    I need help proving this question:

    Suppose lim(n->oo) An = L and let Bn = A2n for all n being Natural numbers. Prove that lim(n->oo) Bn = L
    Use proof by contradiction, suppose that:

    \lim_{n \to \infty} B_n \ne L

    and take it form there.

    CB
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  3. #3
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    More generally, if a sequence converges to L, then every subsequence also converges to L. I personally would be inclined to use a direct proof rather than proof by contradiction. Since \{A_n\} converges to L, given any \epsilon> 0, there exist N such that if n> N, then |A_n- L|< \epsilon. Since B_n= A_{2n}, take n> 2N.

    (If you do use proof by contradiction, note that you cannot assume that \{B_n\} converges. That is, \lim_{n\to \infty} B_n may not exist.)
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  4. #4
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    Yeah my professor said not to use contradiction since there is a much easier way.
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