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  1. #1
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    Question on monotone sequences

    Let A be an infinite subset of the real numbers that is bounded above and let u=supA. Show that there exists an increasing sequence (x_n) with x_n \in A \forall n \in \mathbb{N} such that u=lim(x_n).

    The only way I can think of starting this question is to form some sort of sequence involving u, but can't think of how to do this. Help?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Question on monotone sequences

    Quote Originally Posted by worc3247 View Post
    Let A be an infinite subset of the real numbers that is bounded above and let u=supA. Show that there exists an increasing sequence (x_n) with x_n \in A \forall n \in \mathbb{N} such that u=lim(x_n).

    The only way I can think of starting this question is to form some sort of sequence involving u, but can't think of how to do this. Help?
    Hint:

    Use recursion.

    Maybe I mistaken here, but here it goes:

    Let us defined the following sequence:

    x_{n+1}=\sqrt{u+x_n}, x_1=\sqrt{u}\inA( it because A\neq\emptyset).

    Now, try to prove x_n is monotonic increasing and bounded; and in the last stage find the limit.
    Last edited by Also sprach Zarathustra; August 1st 2011 at 12:17 PM.
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    Super Member girdav's Avatar
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    Re: Question on monotone sequences

    Quote Originally Posted by Also sprach Zarathustra View Post
    x_{n+1}=\sqrt{u+x_n}, x_1=\sqrt{u}\inA( it because A\neq\emptyset).
    How do you know that x_n\in A, \forall n?

    @worc3247: use the definition of \sup: for all \varepsilon>0 we can find x_{\varepsilon}\in A such that u-\varepsilon<x_{\varepsilon}. Now that \varepsilon =\frac 1{n+1} for n\in\mathbb{N}.
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    Re: Question on monotone sequences

    Quote Originally Posted by worc3247 View Post
    Let A be an infinite subset of the real numbers that is bounded above and let u=supA. Show that there exists an increasing sequence (x_n) with x_n \in A \forall n \in \mathbb{N} such that u=lim(x_n).
    That statement is false.
    Let A=[0,1]\cup\{2\}.
    There is no increasing (as opposed to non-decreasing) sequence in A converging to 2.

    (I do not consider a constant sequence as increasing.)
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    Re: Question on monotone sequences

    Well, it is. Otherwise, we have to say strictly increasing.

    By the way, girdav's solution requires the axiom of countable choice. Maybe there is a way without it... I don't see it (except for a set with an ending interval).
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    Re: Question on monotone sequences

    Quote Originally Posted by pece View Post
    Well, it is. Otherwise, we have to say strictly increasing.
    By the way, girdav's solution requires the axiom of countable choice. Maybe there is a way without it... I don't see it (except for a set with an ending interval).
    I beg your pardon. There is a whole school of analysis for which that statement is false. There are just four types of monotone sequences: increasing, decreasing, non-increasing, or non-decreasing. When we see the word 'increasing' it means what it says, the sequence increases. If two consecutive terms are equal then there is no increase is there? So that sequence is possibly non-decreasing but it certainly not increasing.
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    Re: Question on monotone sequences

    Sorry. For us french, increasing means non-decreasing for you. I translate it too literally.

    In that case, it's clearly non-decreasing that would be used.
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    Re: Question on monotone sequences

    The crucial point, as Plato said, is that you cannot prove the original statement- it is false.

    If we are given that the sup of the set, u, is not in the set, then it is true. For any n> 0, there exist a member of the set in the interval from u- 1/n to u- otherwise u-1/n would be an upper bound on the set. Call that number " x_n". The (non-decreasing) sequence \{x_n\} converges to u. A strictly increasing sequence can be derived from that sequence.
    (We need u not in the set since otherwise, as in Plato's example, x_n might be u itself.)

    I have seen textbooks in English that use "non-decreasing; increasing" and others that use "increasing; strictly increasing". Normally which convention is being used is made clear.
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