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Math Help - Proof on supremun

  1. #1
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    Proof on supremun

    Good afternoon.

    I cant figure out this problem. Could someone help.

    Let A be an infinite upper bounded set such that s = supA is not an element of A. Prove that there exists a (strictly) increasing sequence xn in A such that lim(xn) as it approaches infinity is s.

    Thank you. Fares.
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  2. #2
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    Given that s=lub(A) and s is not in A, then s-1 is not an upper bound for A
    Then there is an element a_1 is A and s-1<a_1<s.
    Let e_2=max{s-(1/2),a_1}; so e_2<s.
    Then e_2 is not an upper bound of A, so there is an a_2 in A and e_2<a_2<s.
    If n>2, let e_n=max{s-(1/n),a_n-1}; so e_n<s.
    Then e_n is not an upper bound of A, so there is an a_n in A and e_n<a_n<s.
    Note that a_1<a_2<…<a_n and s-(1/n)<a_n<s.
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