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Math Help - Need help to understand a lemma

  1. #1
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    Need help to understand a lemma

    I need help with understanding Lemma,

    Lemma: For every sequence (an) in R it has monotonic subsequence.

    Firstly, why is this Lemma true?I mean you can have a sequence in R that is monotonically increasing, but I can't find a subsequence that is monotonically decreasing. I can find a subsequence that is monotonic in the same direction but not other directions.. ( By direction I mean increasing and decreasing)

    I think I didn't understand the Lemma can someone explain? Thanks

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  2. #2
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    Re: Need help to understand a lemma

    It doesn't say you need both directions. It just says you need a monotonic subsequence. Clearly if you have a monotonic increasing sequence it's not going to have a monotonic decreasing subsequence and vice versa.
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  3. #3
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    Re: Need help to understand a lemma

    Quote Originally Posted by davidciprut View Post
    Lemma: For every sequence (an) in R it has monotonic subsequence.
    Here is the classic proof.
    Let \mathcal{S}=\{K: (\forall j>K)[a_j>a_K]\}.
    Note that n\in\mathcal{S}\text{ iff }\forall j>n\to~a_j>a_n and n\notin\mathcal{S}\text{ iff }\exists k>n \to a_k\le a_n.

    There are two cases:
    1) \mathcal{S} can be infinite, in which case there is an increasing subsequence.

    2) \mathcal{S} can be finite, in which case there is an non-increasing subsequence.
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