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Math Help - inf(S)

  1. #1
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    inf(S)

    ) Let g : [0; 1] ! R be a continuous function. Let S be a nonempty subset of [0; 1]
    and suppose that g(x) = 3 for all x 2 S. Let a = inf(S). Show that g(a) = 3
    Last edited by mr fantastic; November 27th 2011 at 10:15 AM. Reason: Re-titled.
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  2. #2
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    Re: need question help

    I'm not sure that I understand your question, but if S is a closed set then

    a=\inf(S)=\min(S)

    and since g(x)=3 for all x\in S, we conclude that g(a)=3. If S is an open set you should find a sequence of numbers that converges to a and use the continuity property to prove that \lim\limits_{x_n\rightarrow a} g(x_n)=3 (follow this link)
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  3. #3
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    Re: need question help

    Quote Originally Posted by uasac View Post
    I'm not sure that I understand your question, but if S is a closed set then a=\inf(S)=\min(S)
    and since g(x)=3 for all x\in S, we conclude that g(a)=3. If S is an open set you should find a sequence of numbers that converges to a and use the continuity property to prove that \lim\limits_{x_n\rightarrow a} g(x_n)=3 (follow this link)
    It is not the case that S is either open or closed.
    It is the case that a\in S\text{ or }a\notin S.
    Now this argument works.
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  4. #4
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    Re: need question help

    If S is open then a\notin S.
    If S is closed then a\in S.
    I think...
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  5. #5
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    Re: need question help

    Quote Originally Posted by uasac View Post
    If S is open then a\notin S.
    If S is closed then a\in S.
    I think...
    What you have written there is correct.
    BUT the set S may be neither open nor closed.
    It could be that S=\{0.1\}\cup(0.5,1].
    That set is neither open or closed.
    However, \inf(S)\in S.

    It could be that S=(0.1,0.5]\}\cup\{1\}
    That set is neither open or closed.
    However, \inf(S)\notin S.
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  6. #6
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    Re: need question help

    Right... thanks for the correction.
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