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Math Help - Real Analysis Homework Help!!!!!!

  1. #1
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    Real Analysis Homework Help!!!!!!

    a.)If y>0, then there exists n in N such that n-1(less than or equal to) y<n.
    Prove



    b.)Also: Prove that the n in part (a) is unique...


    Any help would be greatly appreciated!!!!
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  2. #2
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    Quote Originally Posted by trojanlaxx223 View Post
    a.)If y>0, then there exists n in N such that n-1(less than or equal to) y<n.
    Prove



    b.)Also: Prove that the n in part (a) is unique...


    Any help would be greatly appreciated!!!!
    Let A={nεN:n>y}.Now that set is not empty because the Archimedean principal tell us that:

    for all y>0 there exists nεN SUCH THAT n>y

    SINCE A is not empty and also a subset of the natural Nos N, then according to the well ordering principal of natural Nos :

    there exists a kεA AND \forall n. k\leq n.

    But since kεA ,THEN y<k. ALSO k-1<k.

    If now y<k-1 ,then (k-1)εN which implies  k\leq k-1 a contradiction ,hence :  k-1\leq y<k

    For part (b).

    Assume that the set A HAS two minimums k,m both belonging to A

    But since m is a minimum ,  k\leq m...........(1)

    Also since k a minimum ,  m\leq k.............(2)

    From (1) and(2) we conclude k=m ,therefor k is unique

    Do not forget that a minimum of a set is an infemum that belongs to the set.

    If you do not understand part (b) i can give a more detailed proof
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  3. #3
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    Thanks!!!

    Thanks for the help. This question was a part of my review sheet that I couldn't understand. The proof for uniqueness was perfect. Thanks again!!
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