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Math Help - Minimum and Maximum for a convergent sequence

  1. #1
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    Minimum and Maximum for a convergent sequence

    Let (an) be a convergent sequence of real numbers and let A be the set of all its
    terms, i.e.A = {a1, a2, . . . , an, . . . }
    Does A has a minimum or A has a maximum (or both) ?
    Attempt:
    Since the series converges A is bounded .Let s=sup A and i = inf A.
    if s>a and e>0=>|an-s|<e=>|an|<s+e or |an|>s-e .....this implies that there exist and ak belonging to A such that ak>s-e.By def s= max{a1......an0}=>an=<s and s belongs to A.
    The same goes for i=inf A.
    Is this correct?
    Last edited by StefanM; January 3rd 2012 at 08:30 AM.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Minimum and Maximum for a convergent sequence

    Quote Originally Posted by StefanM View Post
    Let (an) be a convergent sequence of real numbers and let A be the set of all its terms, i.e.A = {a1, a2, . . . , an, . . . } Does A has a minimum or A has a maximum (or both) ?
    (i) If a_n=1/n , A has maximum but not minimum. (ii) If a_n=-1/n , A has minimum but not maximum. (iii) If a_1=-1,a_n=1/n\;(n\geq2) , A has maximum and minimum.

    Since the series converges A is bounded .Let s=sup A and i = inf A.
    Right, now prove that if i is not a minimum of A and s is not a maximum of A (hence i<s ) there exists a subsequence with limit i and a subsequence with limit s (contradiction) .
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    Re: Minimum and Maximum for a convergent sequence

    Quote Originally Posted by FernandoRevilla View Post
    (i) If a_n=1/n , A has maximum but not minimum. (ii) If a_n=-1/n , A has minimum but not maximum. (iii) If a_1=-1,a_n=1/n\;(n\geq2) , A has maximum and minimum.
    And, just to be complete, (iv) if a_n= 1- 1/n for n even, a_n= 1/n for n odd, A has neither max nor min.

    Right, now prove that if i is not a minimum of A and s is not a maximum of A (hence i<s ) there exists a subsequence with limit i and a subsequence with limit s (contradiction) .
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Minimum and Maximum for a convergent sequence

    Quote Originally Posted by HallsofIvy View Post
    And, just to be complete, (iv) if a_n= 1- 1/n for n even, a_n= 1/n for n odd, A has neither max nor min.
    Take into account that by hypothesis a_n is convergent.
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