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Math Help - subsequences

  1. #1
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    subsequences

    can someone explain this proof to me?

    theorem:
    let sn be a convergent seq with sn -> l as n tends to infintiy. then every sunseq pf sn converges to l.

    from my notes, it states that by induction, you can easily establise that nk > k for all k....

    sorry, im new to the whole idea of proofs so im not sure how do i use induction to prove that nk > k..

    thanks
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    from my notes, it states that by induction, you can easily establise that nk > k for all k....
    I guess you mean: if (n_k)_{k\geq 0} is a strictly increasing integer-valued sequence, then n_k\geq k for all k\in\mathbb{N}.

    Since n_0\in\mathbb{N}, we have n_0\geq 0, this is the base case.
    Let k\in\mathbb{N}. Assume that n_k\geq k. Let us prove that n_{k+1}\geq k+1. Because (n_k)_k is strictly increasing, n_{k+1}>n_k, hence n_{k+1}>n_k\geq k, and n_{k+1}\in\mathbb{N}, thus n_{k+1}\geq k+1. This concludes the induction.
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